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From: <Saved by Blink>
Snapshot-Content-Location: https://chat.openai.com/c/9b5c4d72-e9fa-43ca-b85a-6136948506c1
Subject: Ambiguous Figures in Philosophy
Date: Sat, 22 Jul 2023 20:45:48 -0000
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<!DOCTYPE html><html class="dark" style="color-scheme: dark;"><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><link rel="stylesheet" type="text/css" href="cid:[email protected]" /><link rel="stylesheet" type="text/css" href="cid:[email protected]" /><link rel="stylesheet" type="text/css" href="cid:[email protected]" /><title>Ambiguous Figures in Philosophy</title><meta name="title" content="ChatGPT: Get instant answers, find inspiration, learn something new"><meta name="keywords" content="ai chat,ai,chap gpt,chat gbt,chat gpt 3,chat gpt login,chat gpt website,chat gpt,chat gtp,chat openai,chat,chatai,chatbot gpt,chatg,chatgpt login,chatgpt,gpt chat,open ai,openai chat,openai chatgpt,openai"><meta property="og:description" content="A conversational AI system that listens, learns, and challenges"><meta name="robots" content="index, follow"><meta name="viewport" content="width=device-width, initial-scale=1"><meta property="og:title" content="ChatGPT"><meta property="og:image" content="https://chat.openai.com/images/chatgpt-share-og.png"><meta property="og:url" content="https://chat.openai.com"><meta name="apple-itunes-app" content="app-id=6448311069"><meta name="next-head-count" content="12"><link rel="apple-touch-icon" sizes="180x180" href="https://chat.openai.com/apple-touch-icon.png"><link rel="icon" type="image/png" sizes="32x32" href="https://chat.openai.com/favicon-32x32.png"><link rel="icon" type="image/png" sizes="16x16" href="https://chat.openai.com/favicon-16x16.png"><link rel="preconnect" href="https://cdn.openai.com/"><link rel="preload" href="https://cdn.openai.com/common/fonts/soehne/soehne-buch.woff2" as="font" crossorigin=""><link rel="preload" href="https://cdn.openai.com/common/fonts/soehne/soehne-halbfett.woff2" as="font" crossorigin=""><link rel="preload" href="https://cdn.openai.com/common/fonts/soehne/soehne-mono-buch.woff2" as="font" crossorigin=""><link rel="preload" href="https://cdn.openai.com/common/fonts/soehne/soehne-mono-halbfett.woff2" as="font" crossorigin=""><link rel="preload" href="https://chat.openai.com/_next/static/css/cc36b64ec67404c4.css" as="style"><link rel="stylesheet" href="https://chat.openai.com/_next/static/css/cc36b64ec67404c4.css" data-n-g=""><meta name="react-scroll-to-bottom:version" content="4.2.0"></head><body class="antialiased"><div id="__next"><div class="overflow-hidden w-full h-full relative flex z-0"><div class="relative flex h-full max-w-full flex-1 overflow-hidden"><div class="flex h-full max-w-full flex-1 flex-col"><div class="sticky top-0 z-10 flex items-center border-b border-white/20 bg-gray-800 pl-1 pt-1 text-gray-200 sm:pl-3 md:hidden"><button type="button" class="-ml-0.5 -mt-0.5 inline-flex h-10 w-10 items-center justify-center rounded-md hover:text-gray-900 focus:outline-none focus:ring-2 focus:ring-inset focus:ring-white dark:hover:text-white"><span class="sr-only">Open sidebar</span><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-6 w-6" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><line x1="3" y1="12" x2="21" y2="12"></line><line x1="3" y1="6" x2="21" y2="6"></line><line x1="3" y1="18" x2="21" y2="18"></line></svg></button><h1 class="flex-1 text-center text-base font-normal">Ambiguous Figures in Philosophy</h1><button type="button" class="px-3"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-6 w-6" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><line x1="12" y1="5" x2="12" y2="19"></line><line x1="5" y1="12" x2="19" y2="12"></line></svg></button></div><main class="relative h-full w-full transition-width flex flex-col overflow-auto items-stretch flex-1"><div class="flex-1 overflow-hidden"><div class="react-scroll-to-bottom--css-mykgk-79elbk h-full dark:bg-gray-800"><div class="react-scroll-to-bottom--css-mykgk-1n7m0yu"><div class="flex flex-col h-full text-sm dark:bg-gray-800"><header class="sticky top-0 z-[9] w-full" style="transform: translateY(-100%); 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Elements in the Philosophy of Mathematics
DOI: 10.1017/9781009039154
First published online: April 2023
Christopher Pincock
The Ohio State University
Author for correspondence: Christopher Pincock, [email protected]
Abstract: This Element answers four questions. Can any traditional
theoryofscientif i cexplanation makesense of the place ofmathematics
in explanation? If traditional monist theories are inadequate, is there
some way to develop a more l exible but still monist approach that will
clarify how mathematics can help to explain? What sort of pluralism
about explanation is best equipped to clarify how mathematics can
helptoexplaininscienceandinmathematics itself?Finally,howcanthe
mathematical elements of an explanation be integrated into the
physical world? Some of the evidence for a novel scientif i c posit may be
traced to the explanatory power that this posit would af f ord, were it to
exist. Can a similar kind of explanatory evidence be provided for the
existence of mathematical objects, and if not, why not?
Keywords: explanation, pluralism, platonism, causation, inference
to the best explanation
© Christopher Pincock 2023
ISBNs: 9781009017664 (PB), 9781009039154 (OC)
ISSNs: 2399-2883 (online), 2514-3808 (print)
Contents
1 Introduction1
2 The Challenge to Traditional Theories of Scientif i c
Explanation2
3 Counterfactual Accounts of Mathematical Explanation19
4 Explanatory Pluralism34
5 Autonomy, Indispensability, and Inference to the Best
Explanation57
References75</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center 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class="w-[30px]"><div class="relative p-1 rounded-sm h-[30px] w-[30px] text-white flex items-center justify-center" style="background-color: rgb(171, 104, 255);"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" stroke-width="1.5" class="h-6 w-6" role="img"><title>ChatGPT</title><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 25.4759 2.66413 26.9026 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10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>It appears you've provided a brief overview and contents of a book named "Mathematics and Explanation: Elements in the Philosophy of Mathematics" by Christopher Pincock. The book seems to tackle some central questions at the intersection of philosophy and mathematics.</p><p>However, without specific questions or a detailed context, it's challenging to provide a comprehensive analysis or overview. Nevertheless, here is a brief understanding of the topics presented in the book:</p><ol><li><p>Introduction: This is likely a summary of the book's main objectives and the central questions it will attempt to answer.</p></li><li><p>The Challenge to Traditional Theories of Scientific Explanation: In this section, the author probably explores how traditional scientific explanations may struggle to incorporate mathematics or mathematical principles effectively.</p></li><li><p>Counterfactual Accounts of Mathematical Explanation: Here, Pincock may present the idea of how different mathematical realities might affect scientific outcomes or realities - in essence, exploring "what if" scenarios in the context of mathematics.</p></li><li><p>Explanatory Pluralism: The author might discuss the idea that there may be multiple, valid ways to explain a scientific concept, each potentially relying on different aspects of mathematics.</p></li><li><p>Autonomy, Indispensability, and Inference to the Best Explanation: This section probably explores the idea that mathematics is both an independent discipline (autonomy) and an essential part of scientific explanation (indispensability). Pincock may discuss the role of mathematics in drawing the best possible conclusion from a set of observations (inference to the best explanation).</p></li></ol><p>Without access to the complete text, this is a speculative outline based on the brief details provided. If you need information on a specific section or question, please provide further details.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">1 Introduction
Some scientif i c explanations involve mathematics. Within mathematics, some
proofs are said to explain. Do these practices tell us anything about the nature of
explanationormathematics?InthisElementthisdauntingtopicisdividedintofour
parts. First, can any traditional theory of scientif i c explanation make sense of the
place of mathematics in explanation? Each traditional theory that is discussed is
a monist theory because it supposes that what makes something a legitimate
explanation is always the same (Section 2). Second, if traditional monist theories
are inadequate, is there some way to develop a more l exible but still monist
approach that will clarify how mathematics can help to explain (Section 3)? After
a consideration of the limitations of some recent l exible monist accounts, the
options for a pluralist approach are examined. What sort of pluralism about
explanation is best equipped to clarify how mathematics can help to explain in
scienceandinmathematicsitself?Whileapluralistcanallowthatdifferentsortsof
explanations work differently, it still remains important to clarify the value of
explanations(Section4).Finally,howcanthemathematicalelementsofanexplan-
ation be integrated into the physical world? Some of the evidence for a novel
scientif i c posit maybe tracedto the explanatory power that this posit would afford,
were it to exist. Can a similar kind of explanatory evidence be provided for the
existence of mathematical objects, and if not, why not? (Section 5).
In his 2001 paper “Mathematical Explanation: Problems and Prospects”
Paolo Mancosu argues that “mathematical explanations can be used to test
theories of scientif i c explanation and that an account of mathematical explan-
ation might have important consequences for the philosophy of science”
(Mancosu 2001, p. 102). This Element builds on this point by considering
how various approaches to scientif i c explanation can make sense of both (i)
explanatory proofs in pure mathematics and (ii) scientif i c explanations that turn
essentially on mathematical resources. In Sections 2–4 I argue that the best
option for clarifying how these explanations work is pluralism about explan-
ation. This means that different explanations employ different explanatory
relevance relations when they indicate why some target is the way that it is. In
Section 5 I consider the signif i cance of mathematical explanation for the
interpretation of pure mathematics. I argue that the existence of genuine math-
ematical explanations does not support the existence of mathematical objects
through the use of inference to the best explanation.
My own interest in mathematical explanation can be traced directly to the
pioneering work of Paolo Mancosu (see especially Mancosu 2000, 2001, 2008,
2018).1I was lucky enough to have Mancosu as the advisor for my 2002
1 A new version of (Mancosu 2018) is currently in preparation.
1Mathematics
and Explanation
https://doi.org/10.1017/9781009039154
Published
online
by Cambridge
University
Press
dissertation on questions related to the applicability of mathematics. This
Element attempts to follow Mancosu’s call to attend carefully to mathematical
and scientif i c practice in philosophical work. I believe that the pluralism about
explanation that I argue for is consistent with Mancosu’s views, but he may not
agreewiththeaccountofwhatallexplanationshaveincommonthatIofferhere
(Section 4.3).</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M11 4H4a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2v-7"></path><path d="M18.5 2.5a2.121 2.121 0 0 1 3 3L12 15l-4 1 1-4 9.5-9.5z"></path></svg></button></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 bg-gray-50 dark:bg-[#444654]"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div 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10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>This introduction provides an overview of the topics that Christopher Pincock intends to address in his book "Mathematics and Explanation: Elements in the Philosophy of Mathematics."</p><p>Here, Pincock lays out four main questions that guide his exploration:</p><ol><li><p>Can any traditional theory of scientific explanation account for the role of mathematics in explanation? He mentions that traditional theories are monistic, meaning they posit a single underlying structure or principle to all explanation.</p></li><li><p>If traditional monist theories fall short, can a more flexible monist approach better elucidate how mathematics contributes to explanation?</p></li><li><p>Pincock then proposes to examine pluralistic approaches to explanation, which accept that different types of explanations may work in fundamentally different ways. He aims to determine which form of pluralism best clarifies the role of mathematics in both science and mathematics itself.</p></li><li><p>Lastly, he wants to consider how mathematical elements of an explanation can be integrated into the physical world.</p></li></ol><p>Pincock refers to the work of Paolo Mancosu, who has argued for the value of examining mathematical explanations in testing theories of scientific explanation and shaping philosophy of science. Inspired by Mancosu, Pincock defends the pluralistic perspective on explanation, arguing that different explanations employ different relevance relations when explaining why something is the way it is.</p><p>However, Pincock distinguishes himself from Mancosu and others who use inference to the best explanation to argue for the existence of mathematical objects. Pincock contends that the existence of mathematical explanations does not necessarily prove the existence of mathematical objects.</p><p>Throughout this work, Pincock seeks to adhere to Mancosu's approach of closely examining mathematical and scientific practice in philosophical investigations, aiming to reveal their inner workings and philosophical implications.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">AdraftofthisElementbenef i ttedenormouslyfromcommentsbySamBaron,
AndreCurtis-Trudel,MarcLange,andPaoloMancosu.Ihaveunfortunatelynot
beenabletoaddressalloftheirhelpfulsuggestionshere,andtheyare,ofcourse,
not responsible for any remaining errors or oversimplif i cations of the issues
discussed. I am also grateful to two anonymous referees for their insightful
reactions to the penultimate version of this Element. I hope this Element will
help to introduce new readers to the wonders of mathematical explanation, and
also to inspire new work on this complex topic.
2 The Challenge to Traditional Theories of Scientif i c Explanation
This section starts by introducing i ve principles that are used to test competing
accounts of explanation, and illustrates how these tests work by developing
some standard objections to accounts that emphasize derivation and unif i cation
(Section 2.1). This section also considers three causal accounts of explanation
and argues that they are unable to make sense of the contrast between explan-
ations that merely employ mathematics to represent something else and explan-
ations whose explanatory power is tied more directly to the mathematics
employed (Section 2.2).
2.1 Derivation and Unif i cation
Philosophical investigations of a topic like explanation typically take for
granted some principles about that topic that make it possible to test competing
accounts. This Element takes for granted i ve principles. It supposes that an
account of explanation aims to cover all explanations, including those found in
science and mathematics. As I will argue, many accounts fail to respect the
principles articulated in this section.We can thus usethese principles toidentify
the problems with various accounts of explanation that have been proposed,
especially when one considers how mathematics i gures into explanations. Of
course, onemay avoidthese problems by rejecting one ormore ofthe principles
that are assumed here.
All participants in these debates agree that a scientif i c explanation provides
a reason why something is the case. The target of an explanation may be
something particular, like a specif i c event or state. The target may also be
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something general, like a recurring pattern or phenomenon. A legitimate
explanation of a target indicates why that target is the way it is. This motivates
our i rst principle for accounts of explanation:
1. There is an important distinction between a description of some target of
explanation and an explanation of that target.
This principle does not def i nitively refute any account, as a defender of any
account is liable to interpret “an important distinction” in their own self-serving
way. However, I will appeal to this principle to help to clarify my reasons for
questioning this or that proposal.2
A closely related principle involves the distinction between the evidence that
some phenomenon has some character and an explanation of that aspect of the
phenomenon. For example, careful paleontological investigations may deter-
mine that the rate of the Earth’s rotation on its axis is decreasing. But additional
accounts of the gravitational interactions between the Earth and the Moon are
needed to explain this change. Our second principle is thus:
2. There is an important distinction between the evidence for some fact and an
explanation of that fact.
The third principle that I will deploy assumes that there is an order to
explanation. If one says that B is the case because of A, then A provides
a reason for B being the way that it is. In some respect, then, A must be more
basic or fundamental than B. In causal explanation, A is partly responsible for
theoccurrenceofB,andsointhissenseisalsomore basic.Ifthisisright,thenit
would be illegitimate to appeal to B when explaining A. One way to summarize
this point is to say that an explanatory relevance relation is asymmetrical: when
A stands in that relation to B, then B does not stand in that very relation to
A. However, the issue is complicated by the fact that an explanation may have
parts. To allow for explanations with parts, our third principle has the following
formulation:
3. (Priority) If A is part of an explanation of B, then B is not part of an
explanation of A.
For example, the mass of the Moon is part of the explanation for why the rate of
rotation of the Earth on its axis is decreasing over time. Our third principle thus
requires that the decreasing rate of rotation of the Earth on its axis is not part of
an explanation of the mass of the Moon.
2 For a recent discussion of this issue, see Taylor (forthcoming).
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Our fourth and i fth principles help to identify the subject matter of this
Element. This is the special character of some explanations that involve math-
ematics. As we will see later in this section, many philosophers maintain that
some explanations that involve mathematics use the mathematics in a special
way that renders the explanation genuinely or distinctively mathematical.
Following Baker and Baron, I will call such explanations “genuine mathemat-
ical explanations.”3Of course, not everyone agrees that there are genuine
mathematical explanations. But our fourth principle takes for granted that
genuine mathematical explanations exist and requires that an account clarify
their character:
4. There is a special way that mathematics may appear in a scientif i c explan-
ation that makes it a genuine mathematical explanation.
Our i fth (and i nal) principle relates to pure mathematics. One goal of
mathematical activity is to obtain a proof of a theorem. Mathematicians some-
times praise or criticize a proof based on its explanatory power. In certain
contexts, it is thought valuable to explain why a theorem is the case even after
it has been given a proof that is otherwise adequate. Our i fth principle assumes
that this feature of mathematical practice is legitimate:
5. Some proofs of a theorem explain why that theorem is the case, while other
proofs do not explain why that theorem is the case.4
Combining our fourth and i fth principles will turn out to be a powerful tool to
criticize some proposed accounts of explanation. Many proposals will fail the
fourth or i fth test because they do not allow for genuine mathematical explan-
ationsortheyruleoutexplanatoryproofs.Aswiththeotherprinciples,thisdoesnot provide a def i nitive refutation of these proposals, but it does clarify their
limitations and also why some may reject those proposals.
Much of our discussion will turn on cases where mathematics appears in an
explanation. Ourf i rstcaseisananswertothequestion,“Whyistheshadowcast
by the Ohio Stadium l agpole 49 m in length at 3 p.m.?” An explanation of this
state may appeal to the position of the sun at the time and to the height of the
l agpole. But this information does not seem suff i cient to explain the length of
the shadow, as there is a deductive gap between the statements that provide this
information and the statement characterizing the target of the explanation:
3 See especially Baker (2005), Lange (2013), and Baron (2019).
4 Contrary to the claims of D’Alessandro (2020), nobody assumes that all explanations in pure
mathematics are proofs. See especially Mancosu (2001) and Lange (2018b). I restrict my focus
here to proofs to make the discussion tractable.
4The Philosophy of Mathematics
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1. At 3 p.m., the light rays from the sun hit the top of the l agpole at an angle of
45°.
2. The height of the l agpole is 49 m.
Therefore, the length of the shadow is 49 m.
To close this deductive gap, we need to add a statement from geometric optics
that involves trigonometry:
3. The length x of the shadow cast by any pole of height y m when the light hits
at an angle of 45° satisf i es the following equation: tan 45° ¼ x m=y m.
As tan 45° = 1 and y = 49, it follows that x = 49 (Figure 1). So for this type of
case, at least, the role for the mathematics in the explanation is to permit the
deduction of a statement characterizing the explanatory target.
Although the idea has a long history, Hempel is the philosopher who did the
most to argue that a necessary condition on an important kind of explanation is
that the explanation provide a deduction of a statement describing the explana-
tory target. Hempel called such explanations “deductive-nomological” (D-N)
explanations. The term “nomological” indicates an additional necessary condi-
tion on such deductions: they must deduce their target statement through the
essential use of a scientif i c law. Our statement 3 would be the law for this
D-N explanation. This is Hempel’s way of distinguishing an explanation from
a description.</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center 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10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>The introduction and part of the second section of this paper focus on the intricate relationship between scientific explanation and mathematics. The paper examines how mathematics can aid explanation in scientific theories. The research divides the topic into four main sections, discussing traditional theories of scientific explanation and their efficacy in incorporating mathematical elements. It discusses whether monistic theories, which believe a single factor legitimizes all explanations, can successfully accommodate the role of mathematics in scientific explanation. The paper proposes a flexible monistic approach that can explicate how mathematics assists in explaining scientific phenomena. The paper then discusses the limitations of these monistic theories and explores potential pluralistic approaches that can better incorporate mathematical explanation. Lastly, the paper discusses the incorporation of mathematical elements into the physical world.</p><p>The author takes inspiration from Paolo Mancosu's work and proposes that pluralism about explanation is the best way to understand how these explanations work. This means that different explanations employ different explanatory relevance relations when they indicate why some target is the way it is. The author argues that the existence of genuine mathematical explanations does not support the existence of mathematical objects through the use of inference to the best explanation.</p><p>The second section discusses the principles used to test different accounts of explanation. The paper talks about five main principles: the difference between a description and an explanation, the difference between evidence for a fact and an explanation of that fact, the priority principle, the presence of genuine mathematical explanations, and the presence of explanatory proofs in mathematics. The paper uses these principles to critique various accounts of explanation that have been proposed, especially those that fail to accommodate the role of mathematics.</p><p>The section also introduces a case that requires the use of mathematics for its explanation, highlighting the necessity and importance of the integration of mathematics into scientific explanations. Finally, it introduces Hempel's concept of "deductive-nomological" explanations, which considers the essential use of a scientific law in the deduction of an explanation, distinguishing it from mere descriptions.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">Hempel offered different motivations for the need for laws in explanations.
In his famous Aspects of Scientif i c Explanation, for example, Hempel (1965)
says “[(i)] the argument shows that, given the particular circumstances and the
laws in question, the occurrence of the phenomenon was to be expected; and
[(ii)] it is in this sense that the explanation enables us to understand why the
phenomenon occurred.” But in addition “[(iii)] it is in virtue of such laws that
the particular facts cited in the explanans possess explanatory relevance to the
explanandum phenomenon” (p. 337).5The relationship between (i), (ii), and
(iii) is far from clear. One interpretation of Hempel is that what makes
something explanatorily relevant is that this fact could have been used to
lawfully predict that state in advance. It is this that constitutes our understand-
ing of that state.
The most inf l uential objection to Hempel’s D-N account takes for granted
that some laws permit deductions with a troubling sort of symmetry. Our
l agpole case was in fact introduced into these debates to illustrate one such
troubling case.6For in addition to the presumably acceptable explanation just
given, the following deductive argument also seems to meet all of Hempel’s
necessary conditions on D-N explanations:
1. At3p.m.,thelightraysfromthesunhitthetopofthef l agpoleatanangleof
45°.
2’. The length of the shadow is 49 m.
3’. Theheightyofanypolethatcastsashadowoflengthxmwhenthelighthits
at an angle of 45° satisf i es the equation: tan 45° ¼ x m=y m.
Therefore, the height of the l agpole is 49 m.
If (3) is a law, then (3’) is also a law. As both arguments are deductively valid,
Hempel seems to lack any principled reason to exclude this explanation. But if
bothexplanationsaregranted,thenweviolateourthirdprincipleconcerningthe
order of explanatory priority. This principle says that if A is part of an explan-
ation for B, then B cannot be part of an explanation for A. But here we have the
height of the l agpole being part of an explanation of the length of the shadow,
and also the length of the shadow being part of an explanation of the height of
the l agpole. Thus we face a choice between agreeing with Hempel and main-
taining an order of explanatory priority. Applying our third principle requires
rejecting Hempel’s approach to explanation.
Most philosophers of science have accepted explanatory priority and thus
rejected Hempel’s D-N account as inadequate. By far the most popular
5 The explanandum is the target of the explanation or what is being explained. The explanans is the
explanation itself or what provides the explanation.
6 See Salmon (1989) for extensive discussion of this and other objections to Hempel.
6The Philosophy of Mathematics
https://doi.org/10.1017/9781009039154
Published
online
by Cambridge
University
Press
approach adds a causal condition on explanation, which we consider in
Section 2.2. However, Kitcher offered a different diagnosis of the failings of
Hempel’s approach. He argued that Hempel failed because he tried to assess
explanations individually. The alternative approach that Kitcher pursued is to
evaluate explanations globally based on how well they help to unify or system-
atize a collection of accepted statements: “Science advances our understanding
of nature by showing us how to derive descriptions of many phenomena, using
the same derivation again and again, and, in demonstrating this, it teaches us
how to reduce the number of types of facts we have to accept as ultimate (or
brute)” (Kitcher 1989, p. 432). At any given time in the history of science, there
will be some set of accepted statements K. The “explanatory store” over K will
specify a set of argument patterns that permit some members of K to be derived
from other members of K. An explanation (with respect to this K) will then be</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex 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relative items-end"><div class="w-[30px]"><div class="relative p-1 rounded-sm h-[30px] w-[30px] text-white flex items-center justify-center" style="background-color: rgb(171, 104, 255);"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" stroke-width="1.5" class="h-6 w-6" role="img"><title>ChatGPT</title><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 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19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>a chain of derivations that uses only patterns in the explanatory store to derive a member of K. This store then represents the unity of scientific knowledge.</p><p>To compare different stores with respect to a given K, Kitcher introduces two measures: the stringency of a store is a measure of its generality, the less stringent the more general; the fertility of a store is a measure of its productivity, the more fertile the more productive. The best store for a given K is then the store that strikes the best balance between stringency and fertility.</p><p>Kitcher’s account of scientific explanation, known as the "unification account," respects our third principle of explanatory priority. In the l agpole case, for example, there is a temporal asymmetry that distinguishes between the height of the l agpole and the length of its shadow. The l agpole exists before the shadow, and it does not change when the shadow does. So, the unification approach would allow the height of the l agpole to explain the length of the shadow, but not vice versa.</p><p>However, Kitcher’s account faces other challenges when it comes to genuine mathematical explanations and explanatory proofs. In both cases, it seems that Kitcher’s approach makes it difficult to accommodate our fourth and fifth principles.</p><p>Take the case of genuine mathematical explanations first. Many examples of such explanations do not fit easily within Kitcher’s unification framework. As an example, let's consider the so-called “number of planets” case, a well-known example of a genuine mathematical explanation. This case involves a mathematical fact, namely that the number 5 is prime, that explains a physical fact, namely that there are no stable solar systems with five celestial bodies following the same orbit. According to Kitcher, an explanation involves deriving a fact from other facts using the patterns in the explanatory store. But in this case, the fact that 5 is prime is not derived from any other facts. Instead, it is the explanatory fact that helps us to derive the physical fact about the solar systems.</p><p>Similar problems arise when we consider explanatory proofs. Many proofs in mathematics are considered explanatory not because they derive a mathematical result from other results in a systematic way, but because they reveal some insight or deep connection that was previously hidden. For instance, a proof of Fermat’s Last Theorem might be considered explanatory if it reveals a deep connection between the theorem and some other areas of mathematics, such as elliptic curves or modular forms. Such a proof would not fit within Kitcher’s unification framework, which focuses on the derivation of facts from other facts in a systematic way.</p><p>In sum, while Kitcher’s unification account of scientific explanation provides an interesting alternative to Hempel’s D-N account, it still faces challenges when it comes to accommodating genuine mathematical explanations and explanatory proofs. These challenges suggest that a different approach may be needed to fully understand the nature of explanation in science and mathematics.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">cases where the mathematics is present in a legitimate explanation but the
mathematics itself is not explanatory. One way to parse such cases is that the
mathematics is there functioning merely to represent something else that is
doingtheexplaining. Butthere might stillbe othercases wherethemathematics
is doing the explaining. In “Mathematical Explanation by Law” Baron calls this
the “genuineness problem”: “to uphold a meaningful distinction between
explanatory and non-explanatory uses of mathematics in science” (Baron
2019).14This is our fourth principle for an account of explanation. Baron’s
plan (in this paper) for how to solve the genuineness problem involves building
on Hempel’s and Kitcher’s emphasis on deduction. But, unlike Hempel and
Kitcher, Baron considers the different ways that mathematics can afford deriv-
ations. In one kind of case, the information that the mathematics contributes
concerning the explanatory target is exhausted by how the mathematics is
representing the target. Consider, for example, a case where a mathematical
claim conveys information about a physical system by i rst picking out an
abstract mathematical structure and then relating it to the target by some kind
of structural mapping. Our l agpole case is like this, where different geometric
lines are interpreted in terms of light rays, physical heights of objects, and
lengths of shadows. For such cases, it looks like we have only a mathematical
representation of some nonmathematical explanation. Baron summarizes this
kind of descriptive information this way: “M contains descriptive physical
information relevant to [target] P when M contains information about an aspect
of some mathematical structure that is mapped into the physical structure
corresponding to P” (Baron 2019, p. 709, emphasis added). But Baron argues
that in other cases the mathematical claims provide information about the target
over and above what is represented via these mappings: in such cases
M provides information about P “in virtue of carrying information about an
aspect of a mathematical structure and yet that aspect is not a relatum in
a mapping relation between the mathematical and physical structures at
issue” (Baron 2019, p. 710).
My worry about Baron’s approach is that information can be provided in this
way, and so not be descriptive in Baron’s sense, and yet still fail to be explana-
tory. To help readers appreciate this worry, I will i rst summarize a case where it
looks as if Baron’s test works. Consider the seven bridges that at one time
connected four areas of the city of Königsberg (Figure 5).
It turns out that it is impossible to make a tour of these bridges that crosses
each bridge exactly once. One explanation of this impossibility relates the
bridges to an abstract mathematical graph (Figure 6).</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M11 4H4a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2v-7"></path><path d="M18.5 2.5a2.121 2.121 0 0 1 3 3L12 15l-4 1 1-4 9.5-9.5z"></path></svg></button></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 bg-gray-50 dark:bg-[#444654]"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative p-1 rounded-sm h-[30px] w-[30px] text-white flex items-center justify-center" style="background-color: rgb(171, 104, 255);"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" stroke-width="1.5" class="h-6 w-6" role="img"><title>ChatGPT</title><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 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10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>In this case, the city of Königsberg is represented by the vertices of the graph and the bridges are represented by the edges. In graph theory, it is a known fact that such a tour, called an Eulerian path, is possible if and only if exactly zero or two vertices have an odd degree (degree refers to the number of edges incident to a vertex). However, in the Königsberg graph, all four vertices have an odd degree, which makes an Eulerian path impossible.</p><p>In Baron's view, the mathematical fact about Eulerian paths is providing information that goes beyond what is represented in the graph. Specifically, the fact that an Eulerian path is impossible is not just a feature of the mathematical structure that represents the Königsberg bridges, but rather an independent mathematical fact that applies to this structure. So, according to Baron, the mathematics is not merely representing the physical situation, but is also contributing to the explanation of why an Eulerian path is impossible.</p><p>However, there might be cases where the mathematical representation provides additional information about a physical situation, yet this information is not explanatory in the relevant sense. Suppose, for example, we have a physical system that can be accurately modeled by a mathematical structure, and suppose this structure has some feature that is not reflected in the physical system. According to Baron's criteria, the mathematics is providing additional information about the system. Yet this information might not be explanatory, if it doesn't illuminate why the physical system behaves as it does.</p><p>To illustrate, consider a situation where we use complex numbers to model a physical system, such as the behavior of an electrical circuit. Complex numbers have a feature, namely their "imaginary" part, that does not correspond to any feature of the physical system. In this case, the mathematical representation is providing additional information (about imaginary parts of complex numbers) that is not mirrored in the physical system. But this information is not explanatory, as it doesn't help us understand why the circuit behaves as it does. This is the worry raised about Baron's approach.</p><p>Thus, while Baron's account offers an interesting way to distinguish between explanatory and non-explanatory uses of mathematics in science, it might not be sufficient to address all cases. The genuineness problem, as Baron calls it, might require a more nuanced understanding of the relationship between mathematics and physical systems, as well as a deeper exploration of what counts as genuinely explanatory information.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">In this section I have considered six strict monist accounts of explanation that
include mathematics. Hempel, Kitcher, and Baron had diff i culties with our i rst
threeprinciplestiedtodescription,evidence,orexplanatory priority.Thecausal
monist views of the new mechanists, Lewis and Woodward made all uses of
mathematics in explanation representational. So, if we insist on our fourth
principle that some mathematical explanations are not representational, then
we must adopt a different approach to explanation.
3 Counterfactual Accounts of Mathematical Explanation
A monist about explanation who wants to include both causal explanations and
genuine mathematical explanations must somehow generalize their explanatory
relevance relation so that it covers both sorts of cases. One approach involves
treating a mathematical claim like a causal law: we consider variations to the
actual world and use the mathematical claim, like we would use a causal law, to
indicate what else would change. This is the core idea of Reutlinger’s general-
ization of a causal counterfactual approach to explanation (Section 3.1).
Another approach treats a mathematical fact as more like a cause: we should
consider variations in the mathematical fact itself and what results from this
variation. This is just like how, in the causal case, we consider what would
happen were a potential cause to be different. I consider three versions of this
approach in Section 3.2. The problems with both approaches to generalizing
a counterfactual approach motivate the discussion of pluralist views in
Section 4.20
3.1 Counterfactuals Involving Only Possible Worlds: Reutlinger
InanumberofpapersReutlingerhasdevelopedamonist“counterfactual theory
of explanation.”21As he puts it, “A monist holds that causal and non-causal
explanations share a feature that makes them explanatory” (Reutlinger 2018,
p. 77). This leads to a theory that takes an especially simple form: something
counts as an explanation just in case it meets four conditions that Reutlinger
calls Structure, Veridicality, Inference, and Dependency. Structure requires that
the explanation include a “nomic generalization” such as a law and some
additional statements relating to initial or boundary conditions. Veridicality
mandates that these statements be true or at least approximately true. To satisfy
Inference, the statements in question must entail the statement to be
20 SeeLange(2021a),KostićandKhalifa(2021),Lange(2022a),andKasirzadeh(forthcoming)for
other discussions of the limitations of counterfactual approaches.
21 Reutlinger (2016), Reutlinger (2017), Reutlinger (2018), and Reutlinger et al. (2022).
19Mathematics
and Explanation
https://doi.org/10.1017/9781009039154
Published
online
by Cambridge
University
Press
explained.22Finally, Dependency requires that the nomic generalization sup-
port a counterfactual with a special form: “if the initial conditions ... had been
different than they actually are (in at least one specif i c way deemed possible in
the light of the nomic generalizations), then E [the target of the explanation] ...
would have been different as well” (Reutlinger 2018, p. 79). A nomic general-
ization supports a counterfactual when the procedure for evaluating the coun-
terfactual leads to the counterfactual being true, on the assumption that the
nomic generalization is true. Reutlinger helpfully illustrates how this works for
a number of examples, including the bridges case we discussed in Section 2.1.
The nomic generalization in this case is the mathematical theorem that a graph
Ghas the right sort of path just in case “(i) all the nodes in Gare connected to an
even number of edges, or (ii) exactly two nodes in G (one of which we take as
our starting point) are connected to an odd number of edges” (Reutlinger 2018,
p. 84). One counterfactual that is supported by this theorem is “if all parts of
Königsberg had been connected to an even number of bridges, then people
would not have failed to cross all of the bridges exactly once” (Reutlinger 2018,
p. 84). The truth of this counterfactual means that Dependency is satisf i ed. As
the other three conditions that Reutlinger imposes are also satisf i ed, the bridges
case counts as an explanation.
Reutlinger’s counterfactual tests for scientif i c explanations do not involve</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M11 4H4a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2v-7"></path><path d="M18.5 2.5a2.121 2.121 0 0 1 3 3L12 15l-4 1 1-4 9.5-9.5z"></path></svg></button></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 bg-gray-50 dark:bg-[#444654]"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative p-1 rounded-sm h-[30px] w-[30px] text-white flex items-center justify-center" style="background-color: rgb(171, 104, 255);"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" stroke-width="1.5" class="h-6 w-6" role="img"><title>ChatGPT</title><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 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19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>Alexander Reutlinger proposes a counterfactual theory of explanation, which attempts to define a single "explanatory relevance relation" that can account for both causal and mathematical explanations. This monistic theory consists of four conditions: Structure, Veridicality, Inference, and Dependency.</p><ol><li>Structure: The explanation must include a nomic generalization (i.e., a law-like statement, which can be a physical law in the case of causal explanations, or a mathematical theorem in the case of mathematical explanations) and some additional statements relating to initial or boundary conditions.</li><li>Veridicality: These statements must be true, or at least approximately so.</li><li>Inference: The statements in the explanation must entail the statement to be explained.</li><li>Dependency: The nomic generalization must support a counterfactual statement of a special form. Specifically, if the initial conditions had been different in some specific way, then the outcome to be explained would also have been different.</li></ol><p>Reutlinger illustrates this approach with the bridges of Königsberg case. The nomic generalization is a mathematical theorem about Eulerian paths, and one counterfactual supported by this theorem is "if all parts of Königsberg had been connected to an even number of bridges, then people would not have failed to cross all of the bridges exactly once." Since all four conditions are satisfied, the explanation counts as an explanation according to Reutlinger's theory.</p><p>However, while this approach can accommodate many cases of mathematical explanation, it's not clear that it can handle all such cases. Some mathematical explanations might not involve counterfactuals in the way that Reutlinger's theory requires. This suggests that a pluralistic approach, which allows for different kinds of explanation in different contexts, might be needed. In particular, we might need to acknowledge that mathematical explanations can sometimes be fundamentally different from causal explanations, and that these differences require different explanatory frameworks.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">mathematical descriptions as explanations or else abandon this counterfactual
theoryin ordertopreserve thedifference between mathematical descriptionand
explanation.
3.2 Counterfactuals Involving Impossible Worlds
Asecondstrategyforgeneralizingacounterfactualanalysisofcausalexplanationis
to suppose that mathematical facts stand in the same relation to their targets that
a cause stands to its effect. With this analogy in mind, we should consider
counterfactuals where that very mathematical fact is altered just as we consider
counterfactuals where the actual cause is altered. This requires counterfactuals that
alter the truth of some mathematical claims. As most assume that these truths are
necessary truths, there will be no possible world as usually conceived to consider.
The three proposals we discuss in this section avoid this by invoking “impossible
worlds” so that these mathematical counterfactuals have nontrivial truth-values.
3.2.1 Baron, Colyvan, and Ripley’s Counterfactual Approach
A generalization of Lewis’ counterfactual account of causal explanation has
been pursued along these lines by Baron, Colyvan, and Ripley (BCR) in two
papers, “How Mathematics Can Make a Difference” (Baron et al. 2017) and “A
Counterfactual Approach to Explanation in Mathematics” (Baron et al. 2020).
The same counterfactual approach is pursued in both articles, but the 2017
article focuses on mathematical explanations of physical phenomena, while the
2020 article discusses mathematical explanations that are internal to mathemat-
ics. I will develop their proposal using two examples that we have discussed so
far: the bridges case and the angle bisection case (Section 2.1). For the bridges,
onekeymathematicalclaimis(M1)thatifagraphhasfournodesandeachnode
has three or i ve edges, then there will not be a sequence of edges that connects
nodes and that includes each edge exactly once. In the angle bisection case,
a central mathematical claim is (M2) that an equilateral triangle can be con-
structed on any line segment. Baron, Colyvan, and Ripley’s proposal is that
these mathematical claims are part of the explanation of their respective targets
just in case a counterfactual with a special character comes out true. The
counterfactual says that were this mathematical claim not to be true, then the
target phenomenon would not occur. That is, were M1 to fail for graphs, then
there would be a circuit of the bridges, and were M2 to fail for line segments,
then there would not be a bisection construction.
These counterfactuals for mathematics raise a number of puzzles concerning
their interpretation and how to evaluate them. Baron, Colyvan, and Ripley
generally follow Lewis in providing a recipe for how to isolate the situation
23Mathematics
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involved: there is an account in terms of similarity between the actual and
nonactual scenarios that pins down what to consider. In a case like the bridges
case, they summarize the procedure as follows:
First, hold i xed the morphism [mapping] between the mathematical structure
S that appears in the counterfactual and the physical structure P. Second, make</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M11 4H4a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2v-7"></path><path d="M18.5 2.5a2.121 2.121 0 0 1 3 3L12 15l-4 1 1-4 9.5-9.5z"></path></svg></button></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 bg-gray-50 dark:bg-[#444654]"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative p-1 rounded-sm h-[30px] w-[30px] text-white flex items-center justify-center" style="background-color: rgb(171, 104, 255);"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" stroke-width="1.5" class="h-6 w-6" role="img"><title>ChatGPT</title><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 25.4759 2.66413 26.9026 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10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>The second strategy for generalizing a counterfactual analysis of causal explanation involves considering mathematical facts in the same way as causal relations. This would involve imagining counterfactual scenarios where mathematical truths are altered, just as we consider counterfactuals where the cause is altered. However, this approach runs into a significant problem: mathematical truths are typically considered to be necessary truths, meaning that they hold in all possible worlds. This means there's no possible world where these truths could be altered.</p><p>To get around this issue, some philosophers propose considering "impossible worlds" – hypothetical situations that violate our usual understanding of logical or mathematical truths. This allows us to give nontrivial truth-values to mathematical counterfactuals.</p><p>Baron, Colyvan, and Ripley (BCR) adopt this approach in their counterfactual account of mathematical explanation. They suggest that a mathematical claim contributes to an explanation if and only if a certain counterfactual is true: if the mathematical claim were not true, then the phenomenon to be explained would not occur.</p><p>However, this approach introduces various puzzles. For example, it's not obvious how to interpret counterfactuals that involve mathematical impossibilities, or how to evaluate their truth. According to BCR, we should compare the actual world to the impossible world by holding fixed the mapping between the mathematical structure and the physical phenomenon, and then altering the mathematical structure in the least disruptive way possible. But even with these guidelines, it's not clear how to proceed in many cases. The concept of an "impossible world" is also quite controversial, and not all philosophers accept its validity.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">privileged mathematical ontology – not just anything goes”
(Povich 2021, p. 537). It does seem right to insist that genuine mathematical
explanations will rely on some privileged mathematical ontology.35However,
Povich’s account mandates ruling out mathematical properties like being near
even simply because they are part of mathematical theorems that have
a biconditional form. By contrast, mathematical practice often takes these prop-
ertiestobesignif i cant,inpart because theydoexplain.Totakea simpleexample,
Euclidean triangles are just those plane i gures whose interior angles sum to 180
degrees. We cannot exclude either of these properties from our mathematical
ontology simply because, if we adopt Povich’s account, it leads to violations of
directionality. Given the ontology that we i nd in mathematical practice, if we
accept directionality, then we must reject Povich’s account.
One could preserve a monist counterfactual account and standard mathemat-
ical ontology by giving up directionality. This is consistent with explanatory
priority, but would involve awkward combinations of explanations that differed
only in the objects involved, as with Königsberg and city L discussed
previously.36 3.3 Conclusion
This chapter has considered four different l exible monist proposals that gener-
alized counterfactual analyses of causal explanation to include some noncausal
cases. The counterfactual theory developed by Reutlinger made do with pos-
sible worlds, but as a result allows for too many noncausal explanations that
violate our principles concerning description, evidence, and priority. Baron,
Colyvan, and Ripley (Baron et al. 2017, 2020; Baron 2020) and Povich (2021)
went further and invoked impossible worlds where some mathematical truths
35 We will return to this point in Section 4.
36 A more thorough investigation of the connections between priority and directionality is not
possible here. See again Kostić and Khalifa (2021) for an analysis of this issue in terms of their
non-ontic approach to topological explanation.
33Mathematics
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https://doi.org/10.1017/9781009039154
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come out false. Here I objected that the accounts either failed to meet our four
principles or, in the case of Povich, failed Povich’s own directionality con-
straint. The upshot of the limitations of these proposals is the rejection of
monism, even in its more l exible form. While some explanations will turn on
the truth of some sort of counterfactual, other explanations will involve
a different explanatory relevance relation. I consider the prospects for this
kind of pluralism in the next section.
4 Explanatory Pluralism
A pluralist about explanation supposes that the best way to make sense of the
variety of explanations in science and mathematics is to posit two or more
explanatory relevance relations. As noted in Section 2, I take these relations to
connect facts in the world when one fact explains another. The views we will
consider in this section accept a notion of causal explanatory relevance. But
eachaddsanewkindofexplanatoryrelevancethatwillaidintheclarif i cationof
genuinelymathematicalexplanations.However,wewillbeginourdiscussionof
pluralist views by considering explanations from pure mathematics. In these
cases some mathematical facts explain another mathematical fact. The pluralist
expansionofrelevancerelationsavoidstheproblemsthatmonistviewsraninto,
butitalsoraisesanewproblem.Isthereanythingthatunitesexplanationsifthey
involve these verydifferentexplanatory relations? In Section 4.3 I will consider
some features that all explanations might have in common, even if pluralism
about explanatory relevance obtains.
4.1 Explanatory Proofs in Mathematics 4.1.1 Steiner on Explanatory Proofs
If we examine mathematical practice, then we i nd that mathematicians value
some proofs because those proofs not only show that a theorem is true, but also
explain why the theorem is true. This is our i fth principle from Section 2.1.
Steiner offered a pioneering analysis of these explanatory proofs.37As we will
seeinSection 4.2.1,Steineralso usedthisanalysis toclarifywhatwe arecalling
genuine mathematical explanations in science. The basic assumption of
Steiner’s approach to explanatory proof is that “to explain the behavior of an
entity, one deduces the behavior from the essence or nature of the entity”
37 Another analysis of explanatory proof was offered by Kitcher and considered in Section 2.1
(Kitcher 1989). See Hafner and Mancosu (2008) for a critical examination of Kitcher’s analysis.
See also Mancosu (2001), Pincock (2015b), D’Alessandro (2021), and Ryan (2021) for add-
itional proposals and cases that are too involved to develop here. One attempt to analyze
explanatory proofs in terms of counterfactuals was brief l y considered in Section 3.2.1.
34The Philosophy of Mathematics
(Steiner 1978a, p. 143). However, Steiner claims that these essences are not
available for mathematical objects. The next best thing is a “characterizing
property” of that object: “a property unique to a given entity or structure within
a family or domain of such entities or structures” (Steiner 1978a, p. 143), where
the notion offamily istakenas basic. We can illustrate characterizing properties
by returning to Euclid’s proof that any angle can be bisected (Section 2.1). As
we saw, this proof can be extended into a family of proofs that any angle can be
divided into n equal parts if, for some m, n = 2m= 2, 4, 8, 16, …. If we are
considering a family of angles divided into equal parts, then a characterizing
property of the objects in this family is how many parts the division has. Then,
one of our proofs shows how a bisected angle is available using a particular
construction, while an angle divided into four equal parts is available using
a closely related construction.
Steiner’s proposal is that an explanatory proof invokes a characterizing
property in a way that allows one to vary that characterizing property to arrive
at other proofs concerning other objects in the domain: “an explanatory proof
depends on a characterizing property of something mentioned in the theorem: if
we ‘deform’ the proof, substituting the characterizing property of a related
entity, we get a related theorem” (Steiner 1978a, p. 147). Deforming a proof
may involve a kind of mechanical manipulation, or it could be something more
creative that preserves the original “proof-idea” (Steiner 1978a, p. 147). Our
collection of angle division proofs satisfy Steiner’s criteria. For we can see how
shifting from a bisected angle to an angle with four equal parts requires an
extended construction. So we have a family of proofs, where appropriate shifts
in the characterizing properties go along with new, successful constructions.
One objection to Steiner’s proposal is that there are nonexplanatory proofs
that meet his tests. The case I will focus on is the classif i cation of i nite simple
groups. A group is a type of object from abstract algebra that has found
application throughout mathematics and physics. The i nite simple groups are
(in a sense) the building blocks of all the other i nite groups, just as the prime
numbers are the building blocks of all the other natural numbers. After a great
deal of effort, an exhaustive classif i cation of the i nite simple groups was
achieved in the 1980s. The resulting proof of the “Classif i cation Theorem”
showed how every i nite simple group was one of 29 identif i able types.38Three
of these types were expected, but the remaining 26 types are called “sporadic”
as they fail to exhibit any clear connections to one another or to the three
expected types. This has led some practitioners to conclude that even though
there is a classif i cation of these groups, we do not yet have an explanation for</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M11 4H4a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2v-7"></path><path d="M18.5 2.5a2.121 2.121 0 0 1 3 3L12 15l-4 1 1-4 9.5-9.5z"></path></svg></button></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 bg-gray-50 dark:bg-[#444654]"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative p-1 rounded-sm h-[30px] w-[30px] text-white flex items-center justify-center" style="background-color: rgb(171, 104, 255);"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" stroke-width="1.5" class="h-6 w-6" role="img"><title>ChatGPT</title><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 25.4759 2.66413 26.9026 2.81182 28.3141C2.95951 29.7256 3.40701 31.0892 4.12437 32.3138C5.18791 34.1659 6.8123 35.6322 8.76321 36.5013C10.7141 37.3704 12.8907 37.5973 14.9789 37.1492C15.9208 38.2107 17.0786 39.0587 18.3747 39.6366C19.6709 40.2144 21.0755 40.5087 22.4946 40.4998C24.6307 40.5054 26.7133 39.8321 28.4418 38.5772C30.1704 37.3223 31.4556 35.5506 32.1119 33.5179C33.5027 33.2332 34.8167 32.6547 35.9659 31.821C37.115 30.9874 38.0728 29.9178 38.7752 28.684C39.8458 26.8371 40.3023 24.6979 40.0789 22.5748C39.8556 20.4517 38.9639 18.4544 37.5324 16.8707ZM22.4978 37.8849C20.7443 37.8874 19.0459 37.2733 17.6994 36.1501C17.7601 36.117 17.8666 36.0586 17.936 36.0161L25.9004 31.4156C26.1003 31.3019 26.2663 31.137 26.3813 30.9378C26.4964 30.7386 26.5563 30.5124 26.5549 30.2825V19.0542L29.9213 20.998C29.9389 21.0068 29.9541 21.0198 29.9656 21.0359C29.977 21.052 29.9842 21.0707 29.9867 21.0902V30.3889C29.9842 32.375 29.1946 34.2791 27.7909 35.6841C26.3872 37.0892 24.4838 37.8806 22.4978 37.8849ZM6.39227 31.0064C5.51397 29.4888 5.19742 27.7107 5.49804 25.9832C5.55718 26.0187 5.66048 26.0818 5.73461 26.1244L13.699 30.7248C13.8975 30.8408 14.1233 30.902 14.3532 30.902C14.583 30.902 14.8088 30.8408 15.0073 30.7248L24.731 25.1103V28.9979C24.7321 29.0177 24.7283 29.0376 24.7199 29.0556C24.7115 29.0736 24.6988 29.0893 24.6829 29.1012L16.6317 33.7497C14.9096 34.7416 12.8643 35.0097 10.9447 34.4954C9.02506 33.9811 7.38785 32.7263 6.39227 31.0064ZM4.29707 13.6194C5.17156 12.0998 6.55279 10.9364 8.19885 10.3327C8.19885 10.4013 8.19491 10.5228 8.19491 10.6071V19.808C8.19351 20.0378 8.25334 20.2638 8.36823 20.4629C8.48312 20.6619 8.64893 20.8267 8.84863 20.9404L18.5723 26.5542L15.206 28.4979C15.1894 28.5089 15.1703 28.5155 15.1505 28.5173C15.1307 28.5191 15.1107 28.516 15.0924 28.5082L7.04046 23.8557C5.32135 22.8601 4.06716 21.2235 3.55289 19.3046C3.03862 17.3858 3.30624 15.3413 4.29707 13.6194ZM31.955 20.0556L22.2312 14.4411L25.5976 12.4981C25.6142 12.4872 25.6333 12.4805 25.6531 12.4787C25.6729 12.4769 25.6928 12.4801 25.7111 12.4879L33.7631 17.1364C34.9967 17.849 36.0017 18.8982 36.6606 20.1613C37.3194 21.4244 37.6047 22.849 37.4832 24.2684C37.3617 25.6878 36.8382 27.0432 35.9743 28.1759C35.1103 29.3086 33.9415 30.1717 32.6047 30.6641C32.6047 30.5947 32.6047 30.4733 32.6047 30.3889V21.188C32.6066 20.9586 32.5474 20.7328 32.4332 20.5338C32.319 20.3348 32.154 20.1698 31.955 20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>This section considers a pluralistic approach to explanations, arguing that making sense of the range of explanations in science and mathematics requires positing multiple explanatory relevance relations. Pluralism proposes that different kinds of explanations are relevant for different kinds of phenomena.</p><p>One example of this approach is found in mathematical practice, where mathematicians often value proofs not only for their logical correctness, but also for their explanatory power - they not only show that a theorem is true, but also explain why it is true.</p><p>Steiner offers a significant analysis of these explanatory proofs. His basic assumption is that to explain the behavior of an entity, one deduces the behavior from the essence or nature of the entity. However, he acknowledges that these essences are not easily available for mathematical objects. Instead, Steiner uses the concept of a "characterizing property," a property unique to an entity or structure within a family or domain of such entities or structures.</p><p>According to Steiner, an explanatory proof depends on a characterizing property of something mentioned in the theorem. If we adjust the proof by substituting the characterizing property of a related entity, we get a related theorem. This idea can be illustrated using angle division proofs.</p><p>However, an objection to Steiner's proposal is that there are nonexplanatory proofs that meet his criteria. An example is the classification of finite simple groups. Although the Classification Theorem categorizes all finite simple groups into 29 identifiable types, the grouping doesn't clearly explain why these groups fall into these categories, particularly for the 26 "sporadic" types that don't exhibit clear connections to each other or to the expected types. As such, even though the Classification Theorem categorizes these groups, it doesn't seem to explain them in a satisfying way.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">4.1.1 Steiner on Explanatory Proofs
If we examine mathematical practice, then we i nd that mathematicians value
some proofs because those proofs not only show that a theorem is true, but also
explain why the theorem is true. This is our i fth principle from Section 2.1.
Steiner offered a pioneering analysis of these explanatory proofs.37As we will
seeinSection 4.2.1,Steineralso usedthisanalysis toclarifywhatwe arecalling
genuine mathematical explanations in science. The basic assumption of
Steiner’s approach to explanatory proof is that “to explain the behavior of an
entity, one deduces the behavior from the essence or nature of the entity”
37 Another analysis of explanatory proof was offered by Kitcher and considered in Section 2.1
(Kitcher 1989). See Hafner and Mancosu (2008) for a critical examination of Kitcher’s analysis.
See also Mancosu (2001), Pincock (2015b), D’Alessandro (2021), and Ryan (2021) for add-
itional proposals and cases that are too involved to develop here. One attempt to analyze
explanatory proofs in terms of counterfactuals was brief l y considered in Section 3.2.1.
34The Philosophy of Mathematics
https://doi.org/10.1017/9781009039154
Published
online
by Cambridge
University
Press
(Steiner 1978a, p. 143). However, Steiner claims that these essences are not
available for mathematical objects. The next best thing is a “characterizing
property” of that object: “a property unique to a given entity or structure within
a family or domain of such entities or structures” (Steiner 1978a, p. 143), where
the notion offamily istakenas basic. We can illustrate characterizing properties
by returning to Euclid’s proof that any angle can be bisected (Section 2.1). As
we saw, this proof can be extended into a family of proofs that any angle can be
divided into n equal parts if, for some m, n = 2m= 2, 4, 8, 16, …. If we are
considering a family of angles divided into equal parts, then a characterizing
property of the objects in this family is how many parts the division has. Then,
one of our proofs shows how a bisected angle is available using a particular
construction, while an angle divided into four equal parts is available using
a closely related construction.
Steiner’s proposal is that an explanatory proof invokes a characterizing
property in a way that allows one to vary that characterizing property to arrive
at other proofs concerning other objects in the domain: “an explanatory proof
depends on a characterizing property of something mentioned in the theorem: if
we ‘deform’ the proof, substituting the characterizing property of a related
entity, we get a related theorem” (Steiner 1978a, p. 147). Deforming a proof
may involve a kind of mechanical manipulation, or it could be something more
creative that preserves the original “proof-idea” (Steiner 1978a, p. 147). Our
collection of angle division proofs satisfy Steiner’s criteria. For we can see how
shifting from a bisected angle to an angle with four equal parts requires an
extended construction. So we have a family of proofs, where appropriate shifts
in the characterizing properties go along with new, successful constructions.
One objection to Steiner’s proposal is that there are nonexplanatory proofs
that meet his tests. The case I will focus on is the classif i cation of i nite simple
groups. A group is a type of object from abstract algebra that has found
application throughout mathematics and physics. The i nite simple groups are
(in a sense) the building blocks of all the other i nite groups, just as the prime
numbers are the building blocks of all the other natural numbers. After a great
deal of effort, an exhaustive classif i cation of the i nite simple groups was
achieved in the 1980s. The resulting proof of the “Classif i cation Theorem”
showed how every i nite simple group was one of 29 identif i able types.38Three
of these types were expected, but the remaining 26 types are called “sporadic”
as they fail to exhibit any clear connections to one another or to the three
expected types. This has led some practitioners to conclude that even though
there is a classif i cation of these groups, we do not yet have an explanation for</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M11 4H4a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2v-7"></path><path d="M18.5 2.5a2.121 2.121 0 0 1 3 3L12 15l-4 1 1-4 9.5-9.5z"></path></svg></button></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 bg-gray-50 dark:bg-[#444654]"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative p-1 rounded-sm h-[30px] w-[30px] text-white flex items-center justify-center" style="background-color: rgb(171, 104, 255);"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" stroke-width="1.5" class="h-6 w-6" role="img"><title>ChatGPT</title><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 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14.4411L25.5976 12.4981C25.6142 12.4872 25.6333 12.4805 25.6531 12.4787C25.6729 12.4769 25.6928 12.4801 25.7111 12.4879L33.7631 17.1364C34.9967 17.849 36.0017 18.8982 36.6606 20.1613C37.3194 21.4244 37.6047 22.849 37.4832 24.2684C37.3617 25.6878 36.8382 27.0432 35.9743 28.1759C35.1103 29.3086 33.9415 30.1717 32.6047 30.6641C32.6047 30.5947 32.6047 30.4733 32.6047 30.3889V21.188C32.6066 20.9586 32.5474 20.7328 32.4332 20.5338C32.319 20.3348 32.154 20.1698 31.955 20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>This section discusses Mark Steiner's work on explanatory proofs in mathematics. Steiner (1978a) presented a pioneering analysis of explanatory proofs, claiming that the explanatory power of a proof lies in the deduction of the behavior of an entity from its essence or nature. However, Steiner posits that essences are not available for mathematical objects. Instead, he suggests the use of a “characterizing property” - a unique property that a given entity or structure within a domain possesses.</p><p>Steiner’s proposal suggests that an explanatory proof hinges on a characterizing property of something mentioned in the theorem, allowing for variations of that characterizing property to derive other proofs concerning other objects in the domain. The "deformation" of a proof by substituting the characterizing property of a related entity may produce a related theorem.</p><p>However, the text raises an objection to Steiner's proposal by presenting a counterexample: the classification of finite simple groups, which was achieved in the 1980s. The exhaustive proof for the classification showed how every finite simple group belonged to one of 29 identifiable types, with 26 of them being sporadic, meaning they don't exhibit any clear connections to one another or to the three expected types. Despite this classification, some mathematicians believe that an explanation for the occurrence and structure of these groups is still lacking. This suggests that there may be non-explanatory proofs that still meet Steiner's criteria, thus posing a potential problem for his proposal.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">4.2 Pluralist Accounts of Genuine Mathematical Explanations
4.2.1 Steiner on Genuine Mathematical Explanations
In a companion paper to the one discussed in Section 4.1.1, Steiner extends his
account of explanatory proofs to an account of what we are calling genuine
mathematical explanations (Steiner 1978b). Recall from Section 2.1 that these
were initially identif i ed as scientif i c explanations that involve mathematics, but
where the mathematics is not functioning merely to represent some nonmathe-
matical explainers like causes. Steiner proposes a simple test for genuine
mathematical explanation: “when we remove the physics, we remain with
a mathematical explanation – of a mathematical truth!” (Steiner 1978b,
p. 19). This can be seen as an ancestor of Baron’s 2019 thought that we should
consider the mathematics that is not used to represent causes or other physical
explainers (see again Section 2.1): we have explanatory information and yet
“thataspectisnotarelatuminamappingrelationbetweenthemathematicaland
physical structures at issue” (Baron 2019, p. 710). Then that explanatory
56 Wussing (2007) is a classic discussion of this history.
46The Philosophy of Mathematics
https://doi.org/10.1017/9781009039154
Published
online
by Cambridge
University
Press
information may extend to associated physical structures, but not simply
because the mathematics is representing causes.
In his 2012 article “Science-Driven Mathematical Explanation,” Baker con-
vincingly argues that Steiner’s account of genuine mathematical explanations
does not work (Baker 2012). Baker invokes genuine mathematical explanations
where no proof is judged to explain the relevant theorem. Baker’s main case
here is the fact that bees use hexagons to build their honeycombs. This is
explained using the theorem that hexagons are the most eff i cient way to tile
a two-dimensional plane while minimizing the tiling materials. While this
explanation seems to be a good one, and also one that qualif i es as a genuine
mathematical explanation, there are no explanatory proofs of the theorem
employed. The same point can made with our example of the bridges of
Königsberg: we showed how this amounted to a genuine explanation of the
impossibility of the bridge crossings, and yet no reference was made to the kind
of proof that the mathematical theorem has.
One could also object to Steiner based on nonexplanations that satisfy all of
his conditions. Consider an uncontroversial case of an explanatory proof from
pure mathematics like the proof of Varignon’s theorem sketched in
Section 4.1.2. The theorem says that for any quadrilateral, if the midpoints of
adjacent sides are connected, then a parallelogram results. Consider any
explanatory target that has the same form as the theorem such as that all metal
wires, when connected to a battery, conduct electricity. Then we can interpret
“quadrilateral” as standing for being a metal wire, “connecting adjacent mid-
points” as standing for being connected to a battery, and “resulting in
a parallelogram” as standing for conducting electricity. This interpretation of
the mathematical theorem then transforms the proof of the theorem into
a derivation of the claim that all metal wires, when connected to a battery,
conduct electricity. But nobody would count this as an explanation. Still,
Steiner’s test is met: “when we remove the physics, we remain with
a mathematical explanation – of a mathematical truth.” So it should be clear
thatthistestisnotagoodone.Oneresponsetothisobjectionistorequireamore
demanding kind of embedding than the simple reinterpretation that I have
sketched here. However, it is not clear how to rule out this sort of problematic
case unless these embeddings are characterized in explanatory terms.</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M11 4H4a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2v-7"></path><path d="M18.5 2.5a2.121 2.121 0 0 1 3 3L12 15l-4 1 1-4 9.5-9.5z"></path></svg></button></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 bg-gray-50 dark:bg-[#444654]"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div 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19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>This section continues the discussion on the philosophy of mathematical explanations, moving onto pluralist accounts of genuine mathematical explanations and revisiting Steiner's theories.</p><p>Steiner, in a follow-up to his previous work, proposes a test for genuine mathematical explanation: if you can remove the physical aspects and still be left with a mathematical explanation of a mathematical truth, then it qualifies as a genuine mathematical explanation. This concept is seen as a precursor to Baron's 2019 idea, which focuses on mathematical aspects that do not function merely to represent physical phenomena.</p><p>However, Baker (2012) contests Steiner's account, pointing out cases where genuine mathematical explanations exist without explanatory proofs for the related theorem. Baker's main example is the honeycomb structure of bees, which uses hexagonal tiling due to its efficiency. This mathematical explanation doesn't rely on a specific proof of the theorem employed.</p><p>The text also points out possible objections to Steiner's proposal, illustrating them with Varignon's theorem, a case from pure mathematics, and its relation to the electrical conductivity of metal wires. Even when Steiner's test is applied and physics is removed, the result doesn't count as a true explanation.</p><p>This shows that Steiner's test may not be a sufficient criterion for a genuine mathematical explanation. Responding to this criticism could require more stringent criteria for the embedding of mathematics within physical phenomena, but characterizing these in explanatory terms poses a challenge.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">5 Autonomy, Indispensability, and Inference to the Best
Explanation
Clarifying how mathematical explanations work is an important part of clarify-
ing how the subject matter of mathematics relates to the subject matter of
sciences like physics, psychology, and economics. In this section I consider
how the relationship between mathematics and physics is both similar to and
different from the relationship between psychology and physics. To start, I will
argue that mathematics is autonomous from physics by comparing mathemat-
ical explanation to psychological explanation (Section 5.1). This is followed by
an examination of some inf l uential arguments for a platonist interpretation of
mathematics (Section 5.2). A platonist claims that mathematics is about
a special domain of abstract objects. An “indispensability” argument for pla-
tonism relies on mathematical explanations having a certain character. I will try
to identify what this character is and then turn to a brief consideration of how an
appeal to inference to the best explanation (IBE) argument could deliver
a platonist interpretation (Section 5.3).
5.1 Putnam on the Autonomy of the Special Sciences
What sort of autonomy should we claim for pure mathematics? One classic
discussion was offered by Putnam. His “Philosophy and Our Mental Life”
argues for a signif i cant analogy between how psychology relates to physics
and how mathematics relates to physics. In the case of both psychology and
mathematics, unlike physics, what is important is that a theory provide
a description of a range of systems that are related by a “functional isomorph-
ism” (Putnam 1975, p. 292). A functional isomorphism obtains when there is
a function that maps the objects, properties, and relations of one system to the
objects, properties, and relations of the other system in such a way that the
structure of the i rst system is preserved in the second system. Putnam develops
this point for mathematics using the example of a cubical peg and a board with
two holes where “the peg passes through the square hole, and it does not pass
through the round hole” (Putnam 1975, p. 295, emphasis removed). Putnam
argues that an explanation of this fact involves the geometrical features of the
peg andthe hole:thecubical peg hasa height of15/16″,the circular holeis 1″in
diameter, and the square has sides that are 1″ long. For Putnam, “The explan-
ation is that the board is rigid, the peg is rigid, and as a matter of geometrical
fact, the round hole is smaller than the peg, the square hole is bigger than the
cross-section of the peg” (Putnam 1975, p. 296). For the peg to i t through the
round hole, the diagonal line connecting two corners of some side of the peg
must be less than the diameter of the hole. But each of these diagonal lines is
about 1.33″ in length (via the Pythagorean theorem). So this is why the peg fails
to go through the round hole. But the square hole is larger than this side of the
square peg, and so it may pass through.
Putnam calls the geometrical explanation of this fact “autonomous”: “The
same explanation will go in any world (whatever the microstructure) in which
those higher-level structural features are present. In that sense this explanation</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M11 4H4a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2v-7"></path><path d="M18.5 2.5a2.121 2.121 0 0 1 3 3L12 15l-4 1 1-4 9.5-9.5z"></path></svg></button></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 bg-gray-50 dark:bg-[#444654]"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative p-1 rounded-sm h-[30px] w-[30px] text-white flex items-center justify-center" style="background-color: rgb(171, 104, 255);"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" stroke-width="1.5" class="h-6 w-6" role="img"><title>ChatGPT</title><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 25.4759 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19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>In this section, the relationship between mathematics and other fields of science such as physics, psychology, and economics is explored. More specifically, it examines the autonomy of mathematics from these sciences and the arguments for a Platonist interpretation of mathematics.</p><p>The autonomy of pure mathematics is compared to the autonomy of psychology in relation to physics, based on a classic discussion by Putnam. In his argument, both psychology and mathematics provide descriptions of systems related by a "functional isomorphism". He illustrates this using the example of a cubical peg and a board with two differently shaped holes, demonstrating that a geometrical explanation of why the peg can pass through one hole and not the other is "autonomous". This explanation remains valid in any world where these higher-level structural features are present, regardless of the microstructure of the world.</p><p>This line of thought introduces the discussion about Platonist interpretations of mathematics, the idea that mathematics is about a special domain of abstract objects. Such a belief often relies on the indispensability argument, which argues that certain mathematical theories are indispensable to our best scientific theories and hence must be true. It then moves towards an evaluation of the appeal to the Inference to the Best Explanation (IBE) argument in order to deliver a Platonist interpretation.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">5.2 Explanatory Indispensability Arguments for Mathematical
Platonism
Many recent discussions of mathematical explanation are motivated by what are
called “enhanced” or explanatory indispensability arguments for platonism.
A platonist says that mathematics is about a special domain of abstract objects.
These objects are outside of space and time and so are causally isolated from
concrete,physicalobjects.ColyvanandBakercanbecreditedwithadvancingthese
newindispensabilityargumentsintheirstrongestform(Colyvan2001,2002,2010,
2012; Baker 2005, 2009b, 2012, 2021). In his 2001 book The Indispensability of
Mathematics,Colyvanpresentedageneralindispensabilityargumentthathetraced
back to Quine and Putnam (Colyvan 2001, p. 11). This argument was then ref i ned
by Baker to focus more directly on explanatory considerations:
1. We ought rationally to believe in the existence of any entity that plays an
indispensable explanatory role in our best scientif i c theories.
64 There is a vast literature on these arguments that tends to focus on psychology. See Lyon (2012)
for an approach to mathematical explanations that uses Jackson and Pettit (1990) to develop an
account of mathematical explanations in these terms.
2. Mathematical objects play an indispensable explanatory role in science.
3. Hence, we ought rationally to believe in the existence of mathematical
objects (Baker 2009b, p. 613).
The entities that an agent ought to believe in are sometimes termed their
“ontological commitments.” In Quine’s inf l uential formulation, encoded in
premise 1, the entities that one takes to exist should be identif i ed through
a regimentation of our best scientif i c theories. In the course of this regimenta-
tion, many apparent ontological commitments will be shown to be not genuine,
as with the average household or shadows. But the remaining ontological
commitments that cannot be eliminated or dispensed with are the ones the
agent is stuck with. Otherwise, in Putnam’s apt formulation, one is embracing
“the intellectual dishonesty of denying the existence of what one daily presup-
poses” (Putnam 2010, p. 57).65
Baker adds a new emphasis on mathematical explanation to this traditional
account of ontological commitment. He can allow the existence of nonmathe-
matical versions of our best scientif i c theories. To show that premise 2 is true,
one need only argue that these nonmathematical versions afford either no
explanation of what the mathematized theory can explain, or else only worse
explanations of what the mathematized theory can explain. Either way, we have
suff i cient reason to include mathematical entities in our ontological commit-
ments. The debate has thus been conducted by focusing on alleged cases of
mathematical explanations of physical phenomena and the potential for non-
mathematical explanations of those phenomena.
Our discussion so far has suggested two different ways of motivating the
explanatory indispensability of mathematical entities. In Section 5.1 I sketched
the argument that mathematics is weakly autonomous with respect to physics.
The weak autonomy of mathematics requires the existence of mathematical
explanations that are better in some respects than any explanation afforded by
physics, for example in terms of generality or robustness. The other approach,
surveyed in Sections 2–4, is to identify a special class of genuine mathematical</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M11 4H4a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2v-7"></path><path d="M18.5 2.5a2.121 2.121 0 0 1 3 3L12 15l-4 1 1-4 9.5-9.5z"></path></svg></button></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 bg-gray-50 dark:bg-[#444654]"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative p-1 rounded-sm h-[30px] w-[30px] text-white flex items-center justify-center" style="background-color: rgb(171, 104, 255);"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" stroke-width="1.5" class="h-6 w-6" role="img"><title>ChatGPT</title><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 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21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>The section starts with an exploration of explanatory indispensability arguments for mathematical Platonism, which is the philosophical view that there exist such things as abstract mathematical objects and these mathematical truths exist independently of humans. Mark Colyvan and Alan Baker have been key figures in promoting this argument.</p><p>Colyvan's work traced back to Quine and Putnam's indispensability argument, which claims that if a mathematical theory is indispensable to a best scientific theory, then we should believe in the mathematical entities postulated by the theory.</p><p>Baker refined this argument to center more directly on explanatory considerations, proposing that if mathematical objects play an indispensable explanatory role in our best scientific theories, then we should rationally believe in the existence of these objects.</p><p>In other words, if we cannot provide a sufficient explanation of the physical world without referencing abstract mathematical objects, then we ought to accept their existence, as they become part of our ontological commitments.</p><p>The discussion thus far presents two ways of motivating the explanatory indispensability of mathematical entities. One is the weak autonomy of mathematics with respect to physics, which argues that mathematical explanations have advantages over physical explanations in terms of generality or robustness. The other approach is to identify a unique class of genuine mathematical explanations.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">ThecontrastsbetweenBaron’sPythagoreanaccountandLange’sAristotelian
realism are instructive. Lange argues that the mappings prized by the platonist
are inadequate to make sense of how the features of abstract mathematical
objects could explain the features of concrete physical objects. So, Lange
concludes that the physical objects must possess special mathematical proper-
ties. At this point, the core difference between these explanatory mathematical
properties and ordinary physical properties is that the mathematical properties
standinnecessaryrelations ofa sortthatisabsentforphysicalproperties.Baron
proceeds in a different direction. He supposes that the mappings prized by the
platonist do involve shared properties between the abstract mathematical
objects and the concrete physical objects. So Baron concludes that at least
some mathematical properties are shared between the abstract and the concrete.
Still,likeLange,forBarontherelationsbetweenthepropertiesarethekeytothe
explanatory power of the mathematics. But Baron claims that the properties are
intrinsic properties of abstract objects, and so they serve to settle the existence
and much of the character of the abstract mathematical objects.
What happens if we combine Baron’s Pythagorean proposal with our super-
venienceprinciple?Baronclaimsthathisaccountisconsistentwithsupervenience:
“It could be that the mathematical properties supervene on physical ones, but it
could also be that some physical properties supervene on mathematical ones. Or
theremaybenosuperveniencerelationatall”(Baronforthcoming,p.11).Thepoint
seems to be that so long as the mathematical properties are distinct from the
physical properties, the existence of genuine mathematical explanations is suff i-
cient for the platonist conclusion. However, Baron is also aware that if super-
venience obtains, “an incompatibility between mathematical properties will
generallycorrespondtoanincompatibilitybetweensubvenientphysicalproperties”
(Baron forthcoming, p. 11) and, we might assume, the same correspondence
obtains for entailments between properties. This seems to undermine the inference
toplatonismforthereasonsnotedinSection5.1:iftherearephysicalpropertiesthat
mimic the explanatory relations provided by the mathematical properties, then the
determined nominalist will regiment their explanations so they mention only the
physicalproperties.SowhileBaron’saccountofgenuinemathematicalexplanation
is consistent with supervenience, supervenience undermines the explanatory indis-
pensability argument that Baron seems to want to defend.
Of course, with Lange’s Aristotelian realist proposal in mind, another obvi-
ous question to ask Baron is why one should posit the existence of abstract
mathematical objects in addition to these intrinsic mathematical properties?
Baron (forthcoming) does not cite Lange (2021b), but Baron does see clearly
how hard it would be for the platonist to use Lange’s analysis of mathematical
explanations as explanations by constraint in a defense of platonism: “how,
exactly, do mathematical objects exert this kind of inf l uence?” (Baron forth-
coming, p. 6). Notice that if we drop the abstract mathematical objects from
Baron’s Pythagorean proposal, we are left with the Aristotelian realist proper-
ties that Lange claims are the subject matter of mathematics. Baron’s response
to this position is direct: “Structural properties on my account make indispens-
able reference to abstract objects” (Baron forthcoming, p. 25). Baron backs up
this assertion by noting some of the diff i culties with developing a nominalist
account of structural properties without in some way or other quantifying over
abstract objects. At this point, though, the debate has shifted away from
questions about genuine mathematical explanation to questions about the non-
explanatory indispensability of mathematics to our best science. If Baron’s</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M11 4H4a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2v-7"></path><path d="M18.5 2.5a2.121 2.121 0 0 1 3 3L12 15l-4 1 1-4 9.5-9.5z"></path></svg></button></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 bg-gray-50 dark:bg-[#444654]"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative p-1 rounded-sm h-[30px] w-[30px] text-white flex items-center justify-center" style="background-color: rgb(171, 104, 255);"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" stroke-width="1.5" class="h-6 w-6" role="img"><title>ChatGPT</title><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 25.4759 2.66413 26.9026 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10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>This section continues the discussion around mathematical Platonism by comparing Baron's Pythagorean account and Lange's Aristotelian realism.</p><p>Lange's argument asserts that abstract mathematical features cannot directly explain concrete physical features. He proposes that physical objects must possess certain special mathematical properties. These mathematical properties are said to stand in necessary relations, a concept absent for physical properties.</p><p>Baron, on the other hand, argues that the abstract mathematical objects and concrete physical objects share certain properties. He contends that these shared properties are intrinsic to the abstract objects, thus determining the existence and character of these abstract mathematical entities.</p><p>There is an exploration of what would happen if Baron’s Pythagorean proposal were combined with the supervenience principle - the idea that some properties are dependent on or determined by others. Baron maintains that mathematical properties could either supervene on physical properties or vice versa, or there may be no supervenience relation at all. However, Baron acknowledges that if supervenience does obtain, it might pose a challenge to the argument for Platonism as there might be physical properties that mimic the explanatory power of the mathematical properties.</p><p>The discussion moves to question why one would posit the existence of abstract mathematical objects in addition to these intrinsic mathematical properties. Lange's Aristotelian realism proposes that we should not posit the existence of abstract mathematical objects. However, Baron claims that structural properties in his account make indispensable reference to abstract objects, hinting at the difficulty in developing a nominalist account of structural properties without quantifying over abstract objects. This, however, shifts the debate away from mathematical explanations and more towards the non-explanatory indispensability of mathematics to our best science.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">5.3 Inference to the Best Mathematical Explanation
Colyvan and Baker have often emphasized that an explanatory indispensability
argument is directed at scientif i c realists. Scientif i c realists typically defend
their realism about unobservable entities like atoms and electrons through the
use of inference to the best explanation (IBE). So, it is claimed, the scientif i c
realist should also accept the existence of platonic mathematical objects using
theverysamesortofIBEargument.AsBakerputsthepoint,“Acrucialplankof
the scientif i c realist position involves inference to the best explanation (IBE) to
justify the postulation in particular cases of unobservable theoretical entities ...
the indispensability debate only gets off the ground if both sides take IBE
seriously, which suggests that explanation is of key importance in this debate”
(Baker 2005, p. 225). On this reading, premise 1 ref l ects an endorsement of an
appropriate form of IBE. Then the existence of the genuine mathematical
explanations required for premise 2 gets us the existence of abstract mathemat-
ical objects. For the very same explanatory considerations that speak in favor of
atoms and electrons also speak in favor of numbers and groups.
It turns out to be diff i cult to articulate and defend a form of IBE that supports
a plausible form of scientif i c realism as well as mathematical platonism. To
appreciate the challenges, consider a classic explanatory argument for the
existence of God popular from the beginning of the nineteenth century. What
explains the existence of the universe and the apparent design that we i nd in
some natural objects like living things? The best available explanation of these
facts in 1800 was that God created the universe and is directly responsible for
these features of living things through various acts of special creation. So, if we
accept IBE, then it looks like we should also accept the existence of God. In
schematic form, IBE is a form of inference that can be boiled down to premise
(A) and conclusion (B):
69Mathematics
and Explanation
https://doi.org/10.1017/9781009039154
Published
online
by Cambridge
University
Press
A. For phenomena P1, ..., Pn, the only available potential explanations are
E1, ..., Er, and of these E1is the best.
B. E1.70
The goodness of the potential explanations is established by considering how
well each explanation exhibits explanatory virtues like simplicity, generality,
and depth. The God explanation was superior in these respects to all other
available explanations, and so it looked reasonable to conclude that God exists.
Itisasimpleexplanationthatrequiresonlyoneentity.Itisageneralexplanation
because it covers a number of phenomena. And it is a deep explanation in the
sensethatitguaranteesthephenomenainquestionbyexplainingalltheirsalient
aspects.71
However, the further development of science raises diff i cult questions about the
useofIBE.Onereadingofthischallengeisthatthevirtuesthatseemedsuff i cientto
activate IBE were actually not suff i cient. That is, scientists who argued for the
existenceofGodin1800usingIBEwereusingitincorrectly.Now,withthebenef i t
of history, we can apply IBE more carefully and in line with a better grasp of the
virtues that are the right ones to appeal to when determining the best explanation.
Consider,f i rst,simplicity.Currently,weexplaintheoriginoftheuniversebyappeal
to the Big Bang and we explain the apparent design shown by many living things
using Darwin’s theory of evolution by natural selection. </div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M11 4H4a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2v-7"></path><path d="M18.5 2.5a2.121 2.121 0 0 1 3 3L12 15l-4 1 1-4 9.5-9.5z"></path></svg></button></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 bg-gray-50 dark:bg-[#444654]"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative p-1 rounded-sm h-[30px] w-[30px] text-white flex items-center justify-center" style="background-color: rgb(171, 104, 255);"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" stroke-width="1.5" class="h-6 w-6" role="img"><title>ChatGPT</title><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 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10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>The section discusses Colyvan and Baker's explanatory indispensability argument, which is directed at scientific realists. Scientific realists often argue for the existence of unobservable entities like atoms and electrons through inference to the best explanation (IBE), a logical reasoning principle used to decide between competing explanations for a phenomenon. Colyvan and Baker argue that the same kind of IBE argument can be used to accept the existence of platonic mathematical objects. This idea suggests that the same reasoning that supports the existence of physical entities like atoms and electrons also supports the existence of mathematical entities like numbers and groups.</p><p>However, the text also points out the challenges in developing a version of IBE that supports both scientific realism and mathematical Platonism. It presents the classical argument for the existence of God as an example: what explains the existence of the universe and the apparent design found in natural phenomena? In the early 19th century, the best explanation was that God created the universe. This argument represents a form of IBE, but further developments in science challenged this conclusion.</p><p>This discussion emphasizes the importance of applying IBE carefully and understanding the virtues that contribute to determining the best explanation, such as simplicity, generality, and depth. Theories that explain the origin of the universe (the Big Bang theory) and the apparent design in many living things (Darwin’s theory of evolution by natural selection) serve as contemporary examples of these explanatory virtues.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">explain the existence of the universe. But for person A, who does not believe in
abstract mathematical objects, the situation isquite different. There isno way to
rationally persuade them to endorse the genuine mathematical explanation
because this explanation will fall below the threshold required by conservative-
ness and modesty. First, in proposing a new sort of object, the mathematical
explanation will be less conservative than its physical competitor. Second, it is
hard to see how the genuine mathematical explanation will satisfy modesty.
Compared at least to the physical explanation, the character of the abstract
objects will not be i xed by what is needed to explain the target physical
phenomena. This is easy to see for a proposed explanation that involves the
natural numbers. It is very unlikely to involve more than a few of the natural
numbers. So the genuine mathematical explanation will not draw on or require
inf i nitely manynumbers. Forthis reason, a competing physical explanation will
be able to get by using only surrogates that simulate whatever i nite structure is
needed to explain the target phenomena. Unlike in the Big Bang case, then, we
get violations of conservativeness and modesty when we try to use IBE to
justify a new belief in the existence of abstract mathematical objects. The
challenge to the platonist is to try to i nd a defensible version of IBE that is
central to a defense of scientif i c realism, while also powerful enough to license
the use of IBE needed to add platonic entities to person A’s ontology. This
challenge has arguably not been met, and this is a way to diagnose why
explanatory indispensability arguments remain unpersuasive.
5.4 Conclusion
Much of the current interest in mathematical explanation is tied to the hope that
a satisfactory account of genuine mathematical explanation would help to settle
the right interpretation of pure mathematics. Our examination of these argu-
ments has been divided into three parts. First, I considered the argument from
Putnam that the special sciences like psychology afford a special kind of
explanation that is not available in physics. I endorsed this argument for what
I called the weak autonomy of mathematics: some mathematical explanations
exhibit explanatory virtues that are absent from their nonmathematical alterna-
tives. Then I turned to explanatory indispensability arguments that try to usethe
special character of genuine mathematical explanations to persuade nonplato-
nists to become platonists. I argued that these arguments do not work. Weak
autonomy is not enough to motivate such a platonist interpretation. The attempt
to supplement weak autonomy with a more detailed account of how mathemat-
ical explanations work also turned out to be insuff i cient for the platonist
conclusion, as becomes clear in Lange (2021b) and Baron (forthcoming).
73Mathematics
and Explanation
https://doi.org/10.1017/9781009039154
Published
online
by Cambridge
University
Press
Finally, we brief l y considered an appeal to IBE in support of platonism. Here
I suggested that a defensible form of IBE should privilege virtues like conser-
vativeness and modesty that restrict when IBE can be used. If the scientif i c
realist only endorses this restricted form of IBE, then they have no reason to
accept the platonist interpretation of mathematical explanations of physical
phenomena.</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M11 4H4a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2v-7"></path><path d="M18.5 2.5a2.121 2.121 0 0 1 3 3L12 15l-4 1 1-4 9.5-9.5z"></path></svg></button></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 bg-gray-50 dark:bg-[#444654]"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div 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10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>The section provides a critical discussion of the use of Inference to the Best Explanation (IBE) to justify belief in abstract mathematical entities.</p><p>It outlines a hypothetical scenario where a person 'A', who does not believe in abstract mathematical objects, cannot be rationally persuaded to endorse a mathematical explanation that introduces a new class of objects. This is due to the fact that such an explanation would violate the principles of conservatism and modesty.</p><p>Conservatism is the principle that an explanation should be as parsimonious as possible, introducing only as many new entities as necessary. Introducing a new category of abstract objects would be less conservative than its physical counterpart.</p><p>Modesty is the principle that an explanation should not go beyond what is required to account for the phenomena. If a mathematical explanation introduces a new class of infinite entities when only a few are required to explain a phenomenon, it would violate this principle.</p><p>As such, the text argues that explanatory indispensability arguments, which aim to convince non-platonists to become platonists using the special nature of mathematical explanations, are not successful.</p><p>The section concludes that the challenge for the platonist is to find a defensible version of IBE that supports scientific realism while also permitting the addition of platonic entities. The text suggests that current attempts to meet this challenge have been unsuccessful.</p><p>Overall, while it is acknowledged that mathematical explanations have a degree of autonomy and can exhibit certain explanatory virtues not found in non-mathematical explanations, the arguments supporting a platonist interpretation of mathematical explanations are found to be unconvincing. The scientific realist, it is argued, has no reason to accept the platonist interpretation of mathematical explanations of physical phenomena.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" 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disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">A summary of the whole article.</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 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9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>I'm sorry, but I don't have the complete text of the article, which means I can't provide a summary of the entire piece. However, based on the sections you've provided, I can summarize as follows:</p><p>The article discusses different perspectives on the role and significance of mathematical explanation in the philosophical debates between Platonism and Nominalism. The text focuses on two primary viewpoints - Baron’s Pythagorean account and Lange’s Aristotelian realism.</p><p>Baron, a Platonist, argues for the existence of shared properties between abstract mathematical objects and concrete physical objects. He posits that these shared properties are intrinsic to the abstract mathematical objects, explaining their existence and character. Baron contends that his approach aligns with supervenience, but acknowledges the risk of it undermining the explanatory indispensability argument he seeks to uphold.</p><p>In contrast, Lange's Aristotelian realism asserts that the physical objects must possess special mathematical properties. He suggests that the existence of these special mathematical properties distinguishes them from ordinary physical properties.</p><p>The text also explores the Inference to the Best Explanation (IBE) argument put forth by Colyvan and Baker. They propose that if scientific realists accept the existence of unobservable entities like atoms and electrons via IBE, they should also accept the existence of Platonic mathematical objects. The article, however, argues that it's challenging to defend an IBE that supports both scientific realism and mathematical Platonism.</p><p>The text concludes by suggesting that explanatory indispensability arguments, aimed at converting non-platonists to platonists, aren't compelling. A defensible form of IBE should adhere to virtues like conservatism and modesty, which restrict when IBE can be used. Therefore, the scientific realist has no reason to accept the Platonist interpretation of mathematical explanations.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">Ok so there is no reason to postulate the existence of an ideal world of forms?</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M11 4H4a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2v-7"></path><path d="M18.5 2.5a2.121 2.121 0 0 1 3 3L12 15l-4 1 1-4 9.5-9.5z"></path></svg></button></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 bg-gray-50 dark:bg-[#444654]"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative p-1 rounded-sm h-[30px] w-[30px] text-white flex items-center justify-center" style="background-color: rgb(171, 104, 255);"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" stroke-width="1.5" class="h-6 w-6" role="img"><title>ChatGPT</title><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 25.4759 2.66413 26.9026 2.81182 28.3141C2.95951 29.7256 3.40701 31.0892 4.12437 32.3138C5.18791 34.1659 6.8123 35.6322 8.76321 36.5013C10.7141 37.3704 12.8907 37.5973 14.9789 37.1492C15.9208 38.2107 17.0786 39.0587 18.3747 39.6366C19.6709 40.2144 21.0755 40.5087 22.4946 40.4998C24.6307 40.5054 26.7133 39.8321 28.4418 38.5772C30.1704 37.3223 31.4556 35.5506 32.1119 33.5179C33.5027 33.2332 34.8167 32.6547 35.9659 31.821C37.115 30.9874 38.0728 29.9178 38.7752 28.684C39.8458 26.8371 40.3023 24.6979 40.0789 22.5748C39.8556 20.4517 38.9639 18.4544 37.5324 16.8707ZM22.4978 37.8849C20.7443 37.8874 19.0459 37.2733 17.6994 36.1501C17.7601 36.117 17.8666 36.0586 17.936 36.0161L25.9004 31.4156C26.1003 31.3019 26.2663 31.137 26.3813 30.9378C26.4964 30.7386 26.5563 30.5124 26.5549 30.2825V19.0542L29.9213 20.998C29.9389 21.0068 29.9541 21.0198 29.9656 21.0359C29.977 21.052 29.9842 21.0707 29.9867 21.0902V30.3889C29.9842 32.375 29.1946 34.2791 27.7909 35.6841C26.3872 37.0892 24.4838 37.8806 22.4978 37.8849ZM6.39227 31.0064C5.51397 29.4888 5.19742 27.7107 5.49804 25.9832C5.55718 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17.849 36.0017 18.8982 36.6606 20.1613C37.3194 21.4244 37.6047 22.849 37.4832 24.2684C37.3617 25.6878 36.8382 27.0432 35.9743 28.1759C35.1103 29.3086 33.9415 30.1717 32.6047 30.6641C32.6047 30.5947 32.6047 30.4733 32.6047 30.3889V21.188C32.6066 20.9586 32.5474 20.7328 32.4332 20.5338C32.319 20.3348 32.154 20.1698 31.955 20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>The text you provided seems to suggest that, at least from certain perspectives in philosophy of mathematics and science, there is no compelling reason to postulate an ideal world of forms, as Platonists would. The argument being that the explanatory indispensability arguments and the inference to the best explanation (IBE) are not sufficient to justify the existence of an abstract world of mathematical objects.</p><p>It's important to note, however, that this doesn't represent a universal consensus. The existence or non-existence of an ideal world of forms is a central topic of debate in philosophy, with various arguments put forward in favor of different positions. Platonists would argue that abstract mathematical objects do exist in a realm of their own, independent of the physical world.</p><p>The article you've provided seems to adopt a more nominalist or Aristotelian realist perspective, which maintains that abstract concepts (like numbers) don't exist as independent entities in some ideal realm but are instead either mere names (in the case of nominalism) or intrinsic properties of physical objects (in Aristotelian realism).</p><p>In the end, the question of whether there's reason to postulate an ideal world of forms is still a matter of ongoing debate and depends on the philosophical perspective one takes.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">Here is the abstract again: Abstract: This Element answers four questions. Can any traditional
theoryofscientif i cexplanation makesense of the place ofmathematics
in explanation? If traditional monist theories are inadequate, is there
some way to develop a more l exible but still monist approach that will
clarify how mathematics can help to explain? What sort of pluralism
about explanation is best equipped to clarify how mathematics can
helptoexplaininscienceandinmathematics itself?Finally,howcanthe
mathematical elements of an explanation be integrated into the
physical world? Some of the evidence for a novel scientif i c posit may be
traced to the explanatory power that this posit would af f ord, were it to
exist. Can a similar kind of explanatory evidence be provided for the
existence of mathematical objects, and if not, why not?</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M11 4H4a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2v-7"></path><path d="M18.5 2.5a2.121 2.121 0 0 1 3 3L12 15l-4 1 1-4 9.5-9.5z"></path></svg></button></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 bg-gray-50 dark:bg-[#444654]"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative p-1 rounded-sm h-[30px] w-[30px] text-white flex items-center justify-center" style="background-color: rgb(171, 104, 255);"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" stroke-width="1.5" class="h-6 w-6" role="img"><title>ChatGPT</title><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 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12.4805 25.6531 12.4787C25.6729 12.4769 25.6928 12.4801 25.7111 12.4879L33.7631 17.1364C34.9967 17.849 36.0017 18.8982 36.6606 20.1613C37.3194 21.4244 37.6047 22.849 37.4832 24.2684C37.3617 25.6878 36.8382 27.0432 35.9743 28.1759C35.1103 29.3086 33.9415 30.1717 32.6047 30.6641C32.6047 30.5947 32.6047 30.4733 32.6047 30.3889V21.188C32.6066 20.9586 32.5474 20.7328 32.4332 20.5338C32.319 20.3348 32.154 20.1698 31.955 20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>Based on this abstract, the full text aims to answer four central questions regarding the role of mathematics in scientific explanation and the possible existence of mathematical objects:</p><ol><li><p><strong>The Adequacy of Traditional Theories:</strong> The paper assesses whether traditional theories of scientific explanation can adequately account for the role of mathematics in explanation. Traditional theories usually posit a singular, unified framework for explanation, and the paper questions whether such an approach can accurately capture the role of mathematics.</p></li><li><p><strong>Possibility of a Monist Approach:</strong> If traditional monist theories (which posit a single kind of explanation) are found inadequate, the text explores if a more flexible monist approach can make sense of how mathematics contributes to explanations.</p></li><li><p><strong>Best Pluralistic Approach:</strong> The text also considers the type of pluralism (the idea that there are multiple kinds of valid explanations) that would best illuminate how mathematics can help explain phenomena in both science and mathematics itself.</p></li><li><p><strong>Integration of Mathematical Elements into the Physical World:</strong> The paper examines how the mathematical elements of an explanation can be integrated into the physical world. This is a significant issue in philosophy of mathematics and science, as it pertains to the relationship between abstract mathematical concepts and the concrete physical reality.</p></li></ol><p>Finally, the abstract suggests that the paper will discuss the issue of explanatory evidence for the existence of mathematical objects. Just as scientific posits (like atoms or particles) might be supported by the explanatory power they offer, the paper will discuss whether a similar kind of evidence could be provided for mathematical objects, and if not, why such evidence is lacking. This relates to the ongoing philosophical debate about the ontological status of mathematical entities - do they "exist" in some abstract realm (as per Platonism), or are they simply useful fictions, conceptual tools, or intrinsic properties of physical entities (as per various other philosophical views)?</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">Similarities with quantum Bayesianism and active inference theories.</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M11 4H4a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2v-7"></path><path d="M18.5 2.5a2.121 2.121 0 0 1 3 3L12 15l-4 1 1-4 9.5-9.5z"></path></svg></button></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 bg-gray-50 dark:bg-[#444654]"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative p-1 rounded-sm h-[30px] w-[30px] text-white flex items-center justify-center" style="background-color: rgb(171, 104, 255);"><svg width="41" height="41" viewBox="0 0 41 41" fill="none" xmlns="http://www.w3.org/2000/svg" stroke-width="1.5" class="h-6 w-6" role="img"><title>ChatGPT</title><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 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20.0556L22.2312 14.4411L25.5976 12.4981C25.6142 12.4872 25.6333 12.4805 25.6531 12.4787C25.6729 12.4769 25.6928 12.4801 25.7111 12.4879L33.7631 17.1364C34.9967 17.849 36.0017 18.8982 36.6606 20.1613C37.3194 21.4244 37.6047 22.849 37.4832 24.2684C37.3617 25.6878 36.8382 27.0432 35.9743 28.1759C35.1103 29.3086 33.9415 30.1717 32.6047 30.6641C32.6047 30.5947 32.6047 30.4733 32.6047 30.3889V21.188C32.6066 20.9586 32.5474 20.7328 32.4332 20.5338C32.319 20.3348 32.154 20.1698 31.955 20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>Quantum Bayesianism (QBism) and active inference theories are concepts that have emerged from different fields, with QBism being a reinterpretation of quantum mechanics, and active inference being a framework in neuroscience and psychology.</p><p>QBism treats the wave function subjectively. It posits that the probabilities in quantum mechanics don't reflect the state of the world, but rather reflect an observer's subjective beliefs about what they can expect from the world. In other words, QBism provides a Bayesian interpretation of quantum mechanics, hence the term "Quantum Bayesianism."</p><p>Active inference, on the other hand, is a theory from neuroscience that suggests that organisms (including humans) continuously infer the state of the world around them and act upon those inferences to minimize surprise and error in their predictions.</p><p>There are a few potential points of overlap or resonance between these theories and the role of mathematics in scientific explanation as discussed in the article you've provided:</p><ol><li><p><strong>Subjectivity and Objectivity:</strong> QBism brings a sense of subjectivity into quantum mechanics, similarly, the role of mathematics in scientific explanations, as discussed in the article, often straddles the boundary between the subjective (mathematics as a language or tool devised by humans) and the objective (mathematics as potentially describing fundamental truths about the universe).</p></li><li><p><strong>Inference and Explanation:</strong> Active inference theories posit that organisms are continuously predicting and updating their understanding of the world. This parallels some of the ideas in the article about how explanations work: we use models (which can be mathematical) to predict and explain phenomena.</p></li><li><p><strong>Pluralism:</strong> The article mentions the possibility of pluralistic approaches to explanation, and both QBism and active inference could be seen as pluralistic in nature. QBism posits a subjective view of quantum mechanics, which could coexist with more objective interpretations. Active inference suggests a view of cognition and behavior that goes beyond simpler stimulus-response models, thus adding another layer to the understanding of brain function.</p></li></ol><p>These are, however, very broad and somewhat speculative connections. The specific similarities or implications would depend on the details of the argument presented in the original article, which has not been fully disclosed in this discussion. Also, it's important to keep in mind that QBism and active inference theories, while fascinating, remain topics of active debate and research within their respective fields.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">Connections with this: arXiv:1201.0541 (quant-ph)
[Submitted on 2 Jan 2012 (v1), last revised 4 Jan 2012 (this version, v2)]
Autangle: A case of Quantum Narcissism?
Rong Zhou, Ryan Orson Behunin, Shih-Yuin Lin, Bei-lok Hu
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In this paper we ask a common psychological question and provide a physics answer: "Looking into a mirror can one get entangled with one's image?" This is not a frivolous question; rather, it bears on the effect of boundaries on the behavior of quantum entanglement between a harmonic oscillator and a quantum field, a basic problem of interest in proposed mirror-field superposition and related experiments in macroscopic quantum phenomena, as well as atomic fluctuation forces near a conducting surface.
The object's internal degree of freedom is modeled by a harmonic oscillator and the presence of a perfectly reflecting mirror enforces the Dirichlet boundary conditions on the quantum field, restricting the latter to a half space. By assuming a bilinear oscillator-field interaction, we derive a coupled set of equations for the oscillator's and the field's Heisenberg operators. The former can be cast in the form of a quantum Langevin equation, where the dissipation and noise kernels respectively correspond to the retarded and Hadamard functions of the free quantum field in half space.
We use the linear entropy as measures of entanglement between the oscillator and the quantum field under mirror reflection, then solve the early-time oscillator-field entanglement dynamics and compare it with that between two inertial oscillators in free space. At late times when the combined system is in a stationary state, we obtain exact expressions for the oscillator's covariance matrix and show that the oscillator-field entanglement decreases as the oscillator moves closer to the mirror. We explain this behavior qualitatively with the help of a mirror image and provide an answer to the question raised above. We also compare this situation with the case of two real oscillators and explain the differences.</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 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21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>The paper you're referencing ("Autangle: A case of Quantum Narcissism?") explores the intriguing question of whether a system can become entangled with its own reflection in a quantum mirror. This research could potentially relate to the discussion on the role of mathematics in scientific explanations in the following ways:</p><ol><li><p><strong>Mathematics as a Descriptive Language</strong>: Quantum mechanics, which the Autangle paper is based on, is a fundamentally mathematical theory. The mathematics not only describes the expected outcomes of quantum systems but also enables counterintuitive phenomena like quantum entanglement. Therefore, the discussion in the Autangle paper is an excellent example of how mathematics is indispensable in providing scientific explanations, especially in complex and abstract domains such as quantum mechanics.</p></li><li><p><strong>Pluralistic Approach to Explanation</strong>: The Autangle paper, like the philosophical text on mathematical explanation, employs a form of pluralism - by exploring different scenarios (oscillator-field interaction vs. two real oscillators) and comparing their outcomes, it offers a richer understanding of quantum entanglement.</p></li><li><p><strong>Inference to the Best Explanation (IBE)</strong>: The Autangle paper uses IBE to arrive at the best explanation for the behavior of an oscillator near a mirror. The mathematical modeling and subsequent comparison of the oscillator's covariance matrix in different scenarios provide an evidence-based explanation of the system's behavior.</p></li></ol><p>Remember that while these connections can provide a basis for conversation and further research, the fields of quantum mechanics, philosophy of mathematics, and mathematical explanation are complex and nuanced, so there is much more to each than what can be covered in a short discussion.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">I.INTRODUCTION
In this note we ask a common psychological question and provide a physics answer: “Looking into a
mirror can one get entangled with one’s image"? According to storybooks the answer for the evil queen
Q seems to be yes, a case of typical royal narcissism.We want to i nd out the answer from physical
considerations: here entanglement refers to quantum entanglement, and autangle means self-entanglement
1 , here referring to the entanglement of a real physical object Q with its mirror image. We hasten to
add that since the image is not a physical object one cannot def i ne, let alone calculate, the entanglement
between a physical object and an unphysical construct. However, there exists entanglement between an
oscillator and a quantum i eld, and we can ask how this is altered if a mirror is present.A perfectly
ref l ecting mirror imposes Dirichlet boundary conditions on the i eld along the mirror surface, restricting its
existence to a half space. These are well-def i ned problems, belonging to the broader inquiry into the ef f ects
of boundaries and topology on the quantum i eld and on quantum entanglement of objects, oscillators and
mirrors, coexisting with and/or mediated by the quantum i eld. In fact, they bear on issues of importance to
quantum information and macroscopic quantum phenomena. Amongst research problems of current interest,
we mention: 1) Entanglement between two two-level (2LA) atoms via a common quantum i eld already shows
diverse quantum entanglement dynamics behavior [2], ranging from sudden death, touch of death, revival
to staying always alive, and features such as dynamical generation, protection, and transfer of entanglement
between subsystems. Model studies of oscillator-f i eld entanglement with experimental ventures have been
carried out [3] as well as oscillator-f i eld entanglement in conjunction with measurements in LIGO detectors
[4], the latter also serving as preparatory studies for mirror-f i eld superposition [5] in macroscopic quantum
phenomena. 2) Entanglement not only changes in time but it also depends on spatial separation as shown in
model studies for 2LAs [6–8] and harmonic oscillators [9] interacting with a common quantum i eld. These
results are of both theoretical interest in understanding what quantum nonlocality means, and practical
value, such as for the design of quantum gates and applications to quantum teleportation. Extending these
i ndings to two and three dimensional systems would enable one to def i ne quantum entanglement domains
[12, 13] and geometric ef f ects of entanglement [14, 15]. The former refers to the ef f ective domain on a surface
or body where entanglement is induced by the presence of an oscillator, not unlike the induced surface charge
density on a conductor or dielectric plate. The latter refers to specif i c geometric patterns in the arrangements
of oscillators where their entanglement strength can be maximized.
Now, returning to the problem under study, one may wonder, is it really necessary to carry out such a
serious calculation in order to answer a simple query posed at the beginning: Is there entanglement between
the queen and her image? The answer is yes, both to the query and to the necessity of a bona i de calculation.
As we cautioned ab initio, an image is not a physical object. Thus we try to avoid invoking an image in
this calculation so that no unphysical assumptions or intuition are brought in unnecessarily or unknowingly
in the derivations. However, at the end, by inspecting the results from the oscillator-f i eld (under mirror
ref l ection) entanglement we see that one can use the notion of an image to describe the process, in fact, come
up with a simple qualitative explanation of how it depends on distance.
The paper is organized as follows: In Section. II we set up the problem and derive a formal set of equations
for the Heisenberg operators of the oscillator’s internal degrees of freedom and the quantum i eld. We impose
the boundary conditions introduced by the presence of the mirror and explain how the commutation relations
are altered. Next we calculate the retarded and Hadamard Green functions of the altered i eld conf i gurations
and derive a quantum Langevin equation for Q(t) including the back-reaction of the altered i eld. We seek
solutions to this equation at late times and calculate the covariance matrix of the oscillator’s canonical
variables at late times when the combined system is in a stationary state. Details are contained in Appendix
A. In Sec.III we introduce the linear entropy and the von-Neumann entropy as measures of quantum
entanglement and calculate the early-time dynamics of entanglement between the oscillator and the i eld in
the oscillator-mirror setup described above. We obtain plots of how the entanglement between the oscillator
and i eld in the half space with the mirror ref l ection evolves with time, depending on the distance between
the oscillator and the mirror. Finally in the discussions section we explain in what sense can one describe
this situation in terms of entanglement of the oscillator with its mirror image and why it decreases as the
oscillator moves closer to the mirror, a somewhat counterintuitive i nding. We also compare this situation
1 We coin this term from two sources of inspiration: new words like 3-tangle and old ones like autism. A more learned word
‘ipso-tangle’ was suggested by Prof. Brill.
3
with the case of two inertial oscillators in free space (calculations placed in Appendix B) and explain their
physical dif f erences.</div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center 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10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>This paper explores the question, "Can one become entangled with one's reflection in a mirror?" from a quantum physics perspective. This inquiry considers entanglement in terms of quantum entanglement, referring to a hypothetical self-entanglement between a real physical object and its mirror image.</p><p>However, since the image isn't a physical object, defining or calculating entanglement between a physical object and a non-physical construct isn't straightforward. Therefore, the paper instead focuses on the entanglement between an oscillator and a quantum field, and how the presence of a mirror impacts this.</p><p>The study pertains to broader investigations into the effects of boundaries and topology on quantum fields and quantum entanglement. The outcomes have implications for quantum information and macroscopic quantum phenomena, including entanglement dynamics, oscillator-field entanglement in experiments like LIGO, and the impact of spatial separation on entanglement.</p><p>In this context, the paper introduces the concept of "quantum entanglement domains" which refers to areas where entanglement is induced by the presence of an oscillator, analogous to the induced surface charge density on a conductor.</p><p>To answer the initial question regarding the queen and her mirror image, the paper outlines a detailed computation process, avoiding the direct involvement of the "image" to prevent any unphysical assumptions.</p><p>The paper then presents calculations of the linear entropy and von Neumann entropy as measures of quantum entanglement, and discusses the dynamics of entanglement between the oscillator and field in the presence of the mirror. The findings suggest a counterintuitive outcome: entanglement decreases as the oscillator moves closer to the mirror. This situation is compared with the case of two inertial oscillators in free space to explain their differences.</p></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 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dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">Simplify it a bit and draw connections to the previous topics.</div></div></div><div class="flex justify-between lg:block"><div 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20.1613C37.3194 21.4244 37.6047 22.849 37.4832 24.2684C37.3617 25.6878 36.8382 27.0432 35.9743 28.1759C35.1103 29.3086 33.9415 30.1717 32.6047 30.6641C32.6047 30.5947 32.6047 30.4733 32.6047 30.3889V21.188C32.6066 20.9586 32.5474 20.7328 32.4332 20.5338C32.319 20.3348 32.154 20.1698 31.955 20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="markdown prose w-full break-words dark:prose-invert light"><p>This study investigates an unusual question from quantum physics perspective: "Can one get entangled with one's own reflection in a mirror?". Here, entanglement is understood as a quantum entanglement. However, because an image in the mirror is not a physical entity, the paper focuses on understanding entanglement between a physical object (an oscillator) and a quantum field in the presence of a mirror.</p><p>This work falls under larger explorations into how boundaries, such as the mirror in this case, affect quantum fields and the entanglement of objects within those fields. The results of these inquiries could have significant implications for quantum information science and larger-scale quantum phenomena.</p><p>The research calculates measures of quantum entanglement - the linear entropy and von Neumann entropy - and investigates how entanglement changes over time and with varying distances between the oscillator and the mirror. Interestingly, the findings suggest that entanglement decreases as the oscillator gets closer to the mirror.</p><p>Connections to Previous Topics:</p><ol><li><p>With regard to the discussion on mathematical explanation in science, this study is an example of how mathematical and physical concepts (in this case, quantum entanglement and boundary conditions) are used to explain and predict physical phenomena.</p></li><li><p>As for Quantum Bayesianism and Active Inference theories, these are frameworks that interpret quantum mechanics and cognitive processes respectively. This study on mirror entanglement is in line with Quantum Bayesianism in terms of how it uses probabilistic reasoning (entropy measures of entanglement) to describe quantum states.</p></li><li><p>It can also be related to Active Inference theory, as it is another example of how systems (in this case, an oscillator and a quantum field) interact with their environment (influenced by a mirror) and adapt based on those interactions (change in quantum entanglement).</p></li></ol></div></div></div><div class="flex justify-between lg:block"><div class="text-xs flex items-center justify-center gap-1 self-center pt-2 !invisible"><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0">1 / 1</span><button disabled="" class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="1.5" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-3 w-3" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect x="8" y="2" width="8" height="4" rx="1" ry="1"></rect></svg></button><div class="flex gap-1"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400"><svg stroke="currentColor" fill="none" stroke-width="2" viewBox="0 0 24 24" stroke-linecap="round" stroke-linejoin="round" class="h-4 w-4" height="1em" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div><div class="group w-full text-gray-800 dark:text-gray-100 border-b border-black/10 dark:border-gray-900/50 dark:bg-gray-800"><div class="flex p-4 gap-4 text-base md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl md:py-6 lg:px-0 m-auto"><div class="flex-shrink-0 flex flex-col relative items-end"><div class="w-[30px]"><div class="relative flex"><img alt="User" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="rounded-sm" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F6241bfe48785687ad5d68f5c836fbe6d%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fna.png&w=96&q=75" style="color: transparent;"></div></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]"><div class="flex flex-grow flex-col gap-3"><div class="min-h-[20px] flex items-start overflow-x-auto whitespace-pre-wrap break-words flex-col gap-4"><div class="empty:hidden">II.OSCILLATOR INTERACTING WITH FIELD UNDER MIRROR BOUNDARY CONDITION
A.Dynamics of Oscillator-Field System with Mirror
As described in the Introduction, the gist of the matter for this problem is to quantify the change in
some entanglement measure between the oscillator and the i eld with and without the presence of a mirror.
Technically our calculation involves two parts: 1) i nd the back-reaction of the i eld conf i guration altered by